J. Phys. Chem. 1988, 92, 2123-2128
2723
that the wave packets initially centered in a regular region of phase space will slowly disperse while following the periodic classical trajectory. For chaotic regions of phase space the wave packet will disperse much more rapidly. Thus even though He is trapped in the valleys for a relatively short time, it turns out that this time is long enough to mark the quantum behavior. The association of the characteristics of classical trajectories with the degree of rotational excitation in collisions between H e and LiH has been examined by Metropoulos and Silver.21 By analyzing the torque, they have shown that large AN are favored from trajectories that are mainly of direct type. However we point out that their potential is not as anisotropic as HeOH.
J
Figure 11. Partial cross sections for rotational excitation in closecoupling calculations at E = 0.05 eV and for two different reduced masses.
deeper in the valleys of the potential, and this explains the decrease of the the cross sections of even transitions. Furthermore, in Figure 11 the partial cross sections for p1 = 6.5 amu show secondary maxima not only for the 1-+2 transition as in Figure 3 but for the 0-+2 and 1-3 transitions as well. On the other hand, at p l = 1.5 amu all the trajectories are direct, and the partial cross sections have no secondary maxima (Figure 11). Grimes et al.' have also found multiple maxima in the partial cross sections in He + H2(B1Z,+) rotationally inelastic collisions, which disappeared when the potential was modified in order to be dominated by the Voterm, i.e., the tential became less wavy. They conclude that the maxima in OJO2.-o are due to the dominance of the V, term for 0.6 IR I12.0~0. The correspondence that we establish between quantum and classical mechanics resembles the results that have been obtained for bound systems. Heller19 has shown that the wave functions in the stadium model show an increased amplitude across the unstable classical periodic orbits. Moiseyev and PeresZohave found (19) Heller, E. J. Phys. Rev. Lett. 1984, 53, 1515. (20) Moiseyev, N.; Peres, A. J . Chem. Phys. 1983, 79, 5945.
Conclusions Close-coupling but spin-averaged total cross sections for rotational excitations in collision of He with OH(A22+) have been calculated on an ab initio potential energy surface. The potential is repulsive, but it shows minimum energy paths of approach of He to O H at 6 = 45' and 135'. Because of interference effects the cross sections for even transitions, u ~ +and~ uqc2, are found to be larger than the uZc1and u3.+ respectively. These inversions are not reproduced in the coupled-states approximation. The magnitudes of the cross sections are comparable to those found in similar systems but for ground-state potentials. As an example we mention the work of Battaglia et a1.22on He + HF collisions. The partial cross sections show a structure with secondary maxima that varies with the collision energy. This structure is also present in the C S partial cross sections, which means that the maxima do not result from interference effects among the partial waves. Thus a classical analogue can be found. Indeed analysis of classical trajectories of collisions of He with OH reveals two types of trajectories that promote a particular transition: direct, which show a linear time evolution of the angular momenta phases, and trapped trajectories, which show an oscillatory behavior of the phases. When the transitions are promoted by both types of trajectories, the quantum partial cross sections show secondary maxima. For energies at which a transition occurs only from one type, the partial cross section has one maximum. Acknowledgment. We are grateful to Dr. Hutson for the provision of the MOLSCAT programs. Registry No. He, 7440-59-7; OH, 3352-57-6. (21) Metropoulos, A.; Silver, D. N. Chem. Phys. Lerr. 1982, 93, 247. (22) Battaglia, F.; Gianturco, F. A,; Palna, A. Chem. Phys. 1984,84, 233.
Ab Initio Potential Energy Surfaces of He(%) 4- OH(X2n) and He(%)
+ OH(A22+)
A. Vegirit and S. C. Farantos* Department of Chemistry, University of Crete, and Institute of Electronic Structure and Laser, Research Center of Crete, 711 10 Iraklion. Crete, Greece (Received: May 4, 1987)
Ab initio potential curves of H e 0 and potential energy surfaces of HeOH for the first three states are presented. The potential surfaces are found to be repulsive with the following topographical characteristics: (1) The minimum energy approach of He to hydroxyl radical is across the angles of 45' and 135'. (2) There is no strong coupling between the bond length of OH and the distance of He from the center of mass of the diatom, R. (3) There is a surface crossing between the 211 and 2Z+state as He collinearly approaches the 0 atom of OH at R = 2.3 a,, and at 2.0 eV above the dissociation limit, He + OH(A*Z+). An analytical potential function is produced for the excited Z state.
1. Introduction Ab initio potential energy surfaces (pes) have been produced for a few triatomic systems and mainly for the ground electronic state.{ On the other hand, experimental information for the 'Also Department of Physics, University of Crete.
0022-3654/88/2092-2723$01.50/0
excited electronic states of small molecules has been gathered recently.'-' These results put forward questions related to the (1) Tully, J. C. In Dynamics of Molecular Collisions, Part B; Miller, W . H., Ed.;Plenum: New York, 1976; Chapter 5, p 217. (2) Smalley, R. E. Annu. Rev. Phys. Chem. 1983, 34, 129.
0 1988 American Chemical Society
2724
The Journal of Physical Chemistry, Vol. 92, No. 10, 1988
dynamics of collisions of an atom with an electronically excited molecule. Thus it is necessary to calculate potential energy surfaces accurate enough to resolve such questions. A recent study of He('S) H2(B'2,+)6-10has demonstrated that knowledge of the complete pes can explain the observed behavior of this ~ y s t e m . ~ In this article we present ab initio MRD-CI calculations" for He(%) colliding with O H in the ground X211 and excited A2Z+ states. OH plays an important role in combustion and atmospheric chemistry.12 In interstellar space it has been found that hydroxyl radical shows a maser action involving transitions across the ground-state A-doublet population i n ~ e r s i o n . ' ~ Schinke and AndresenI4 have recently studied inelastic collisions of OH(X211) with H, by using quantum mechanical scattering theory. The results are favorably compared with the observed values. OH in the excited A22+state colliding with He has been studied by Lengel and C r ~ s l e y .These ~ authors measured electronic and vibrational quenching rates for the vibrational states u = 1 and 2 as a function of the rotational quantum number of OH(AZZ+) in room-temperature collisions with several partners. They found that H e and Ar are not so efficient quenchers as the diatomics H,, D2, and NZ.These studies have been extended by Crosley and co-workers15as well as by Papagiannakopoulos and Fotakis.'6 The latter authors have measured state-specific rate constants for the deactivation of OH excited at high rotational states by H 2 0 and CO. They conclude that the mechanism of deactivation depends on the nature of the collisional partner. From the above it is clear that there is an increasing interest in quantum chemistry calculations of systems that involve O H in either the ground or excited state. Furthermore, the production of pes for electronically excited states will allow us to examine the correlation of surface topography and dynamics when the channel of electronic quenching is open. The outline of this paper is as follows: In section 2 we give the computational details, In section 3 we present the MRD-CI calculations for the diatom He 0 in the ground (X311) and the first two excited potential curves 32-and l Z f as well as the computations for the first three surfaces of He + OH. An analytical potential energy surface is presented for the excited state of HeOH. In section 4 we discuss the implication of the surface topography to collision dynamics whereas in the companion paper in this issue we give the results of a quantum mechanical calculation for rotational excitation on the H e + OH('Z+) excited surface.
+
+
2. Computational Details Dissociation Limits. The double degeneracy of the X211 state (3) Akins, D. L.; Fink, E. H.; Moore, C. B. J . Chem. Phys. 1970,52, 1604. Fink, E. H.; Akins, D. L.; Moore, C. B. J . Chem. Phys. 1972, 56, 900. (4) Comes, F. J.; Fink, E. H. Chem. Phys. Lett. 1972, 14, 433m. Fink, E. H.; Comes, F. J. Chem. Phys. Lett. 1974, 25, 190. (5) Lengel, R. K.; Crosley, D. R. J . Chem. Phys. 1978, 68, 5309. (6) Farantos, S. C.; Theodorakopoulos, G.; Nikolaides, C. A. Chem. Phys. Lett. 1983. 100. 263. (7)Thebdorakopoulos, G.; Farantos, S.C.; Buenker, R. J.; Peyerimhoff, S. D. J . Phys. B 1984, 17, 1453. (8) Knowles, D. B.; Murrell, J. N.; Braga, J. P. Chem. Phys. Lett. 1984, 110, 40.
(9) Farantos, S.C. Mol. Phys. 1985, 54, 835. 1986, 59, 1273. (10) Grimes, R. M.; Lester, W. A,; Dupuis, M. J . Chem. Phys. 1986, 84, 5437. (1 1) (a) Buenker, R. J.; Peyerimhoff, S. D. Theor. Chim. Acta 1974,35, 33. 1975, 39, 217. Buenker, R. J.; Peyerimhoff, S. D.; Butscher, W. Mol. Phys. 1978, 35, 771. (b) Buenker, R. J. In Proceedings of Workshop on Quantum Chemistry and Molecular Physics; Wollongong, Australia, 1980. Buenker, R. J. Stud. Phys. Theor. Chem. 1982, 21, 17-34. Buenker, R. J.; Phillips, R. A. J . Mol. Struct. 1985, 123, 291. (12) Levy, H. I1 Science 1971, 173, 141. Crosley, D. R. Opt. Eng. 1981, 20, 511. Kleinermanns, K.; Wolfrum, J. Laser Chem. 1983, 2, 339. (13) Gwinn, W. D.; Turner, B. E.; Goss, W. M.; Blackman, G. L. Asrrophys. J . 1973, 179, 789. Bertojo, M.; Cheung, A. C.; Townes, C. H. Ibid. 1976, 208, 914. Shapiro, M.; Kaplan, H. J . Chem. Phys. 1979, 71, 2182. Dixon, R. N.; Field, D. Proc. R. SOC.London, A 1979, 368, 99. (14) Schinke, R.; Andresen, P. J . Chem. Phys. 1984, 81, 5644. (15) Fairchild, P. W.; Smith, G. P.; Crosley, D. R. J . Chem. Phys. 1983, 79, 1795. (16) Papagiannakopoulos, P.; Fotakis, C. J . Phys. Chem. 1985, 89, 3439.
Vegiri and Farantos TABLE I: Comparison of the Total Energies (hartree) Given by Two Different Basis Sets for the He + OH System' state R = 2.5 a,, R = 3.0 an R = 10.0 a" 1A'
-78.431 56 -78.462 56 -78.348 17 -78.380 04 -78.19583 -78.22847 -78.131 99 -78.151 95
1A" 2A' 2A"
-78.484 65 -78.51276 -78.450 18 -78.482 88 -78.296 70 -78.326 89 -78.18621 -78.19460
-78.51357 -78.544 59 -78.51364 -78.545 10 -78.357 42 -78.390 64 -78.220 50 -78.234 17
"Upper numbers are obtained with basis set 111 of Langhoff et al.'* Lower numbers are given with basis set used in this work.28 The bond length of OH is 1.83 a,,, and 8 = 90°.
of OH is removed when He approaches the molecule across C, geometries. Two surfaces then emerge, the 1*A' and 12A", which are intersected at collinear geometries. This is a typical Renner-Teller inter~ection.'~The dissociation limits of these states are given in (1-5). HeOH(12A', 12A")
-
-
-
-
O(3P)
+ H(,S) + He('S)
(1)
OH(XZII) + He('S)
HeH(X*Z+) + O(3P)
HeO(X311) + H('S)
HeO(A'2')
+ H(%)
R H ~> O 2.5
00
RHeO 2.5 a.
(4)
(5)
Ab initio calculations for the construction of diatomic potential curves have been performed for OH'8-23and HeH.7j8 On the other hand, the helium oxide has not been studied. Only recently Staemmler and JaquetZ6carried out CEPA calculations to determine the long-range part of the O(3P) + He('S) interaction. Potential curves for the 0(3P,'D,'S) interacting with Ne, Ar, Kr, and Xe have been produced to understand processes connected with possible laser action in these systems.27 Thus our determination of the repulsive part of helium oxide potential is complementary to the above studies. In section 3 it is shown that channels 4 and 5, as the minimum energy dissociation limits, are dictated by a crossing of the two curves X211 and A2Z+. The excited state has the dissociation channels given in (6-10).
-
-
-
+ H(%) + He('S) OH(A2.Z+) + He('S) HeH(XZZ+)+ O('D) HeO(A'Z+) + H(%) RHeO > 2.5 a. HeO(X311) + H(%) RHeO< 2.5 a.
HeOH(2,A')
-
O('D)
(6) (7)
(8) (9)
(10)
(17) Teller, E. J. Phys. Chem. 1937, 41, 109. Herzberg, G. H.; Lonpet-Higgins, H. C. Discuss. Faraday SOC.1963, 35, 77. (18) Langhoff, S.R.; van Dishoeck, E. F.; Wetmore, R.: Dalgarno, A. J . Chem. Phys. 1982, 77, 1379. (19) van Dishoeck, E. F.; Langhoff, S.R.; Dalgarno, A. J . Chem. Phys. 1983, 78,4552. van Dishoeck, E. F.; Dalgarno, A. J . Chem. Phys. 1983, 79, 873. (20) Chu, S I . ; Yoshimine, M.; Liu, B. J . Chem. Phys. 1974, 61, 5389. (21) Stevens, W. J.; Das, G.; Wahl, A. C.; Neumann, D.; Krauss, M. J . Chem. Phys. 1974,61, 3686. (22) Werner, H.-J.; Rosmus, P.; Reinsah, E.-A. J . Chem. Phys. 1983, 79, 905. (23) Wright, J. S.; Barclay, V.; Kranus, E. Chem. Phys. Lett. 1985, 122, 214. (24) Moore, C. Natl. Bur. Stand. (US.) Circ. No. 467 1949, 1; 1952, 2; 1958, 3. (25) Romelt, J.; Peyerimhoff, S.D.; Buenker, R. J. Chem. Phys. 1978,34, 403. (26) (a) Staemmler, V.; Jaquet, R. Chem. Phys. 1985,92, 141. (b) Jaquet, R.; Staemmier, V. Chem. Phys. 1986, 101, 243. (27) Dunning, T. H., Jr.; Hay, P. J. J . Chem. Phys. 1977, 66, 3767.
He(lS)
+ OH(X211) and He(IS) + OH(A2Z+)
TABLE II: Comparison of the OH Energies (electronvolt) Calculated with He at 10 a n r/ao state CI' F C I ~ CYLC LDWD(III)~ LDWD(1)' 1.35 1.184 1.5 211 1.145 1.177 1.158 5.74 5.549 5.650 5.587 5.629 2z+ 0.0 0.0 0.0 1.83 21T 0.0 0.0 4.199 4.148 4.187 4.34 4.063 2z+ 0.144 0.193 0.155 0.08 2.0 2n 4.210 4.30 4.081 2z+ 4.310 1.29 1.311 1.455 1.414 1.445 2.5 211 4.99 4.820 2z+ 5.157 5.036 5.024 2.725 2.698 2.81 1 2.56 2.659 3.0 211 5.79 5.654 6.01 1 5.867 5.859 2z+ 3.39 3.540 3.5 2n 3.507 3.525 3.637 6.10 6.1 11 2z+ 6.303 6.294 6.264
'Extrapolated values." The absolute value of the zero energy is -78.54510 hartree. bFull CI. CFromref 20. dFrom ref 18 obtained with the Gaussian basis set 111. eFrom ref 18 obtained with Slater type basis set I. Dissociation channel 7 is the lowest energy path for this surface when the OH bond length, r, is less than 2.5 ao. Ab initio calculations on the hydroxyl radicalIg have shown that the repulsive 122-curve, which dissociates to ground-state atoms, crosses the AZZ+potential at a distance of 2.5 ao. Therefore we expect the 22A" potential surface to intersect the 22A' surface at this value of the OH bond length. The 22A" state correlates with the %of OH and 3 X - of HeO. Although this potential surface is estimated in our calculations as the second root of A" symmetry, it is not of the same accuracy as the lower energy states, and therefore it is not presented in this paper. Numerical Methods. The calculations were performed on a VAX 11/750 computer with the MRD C I programs developed by Buenker and Peyerimhoff." The atomic basis functions were selected from the literature. The particular basis set we usez8 is a contracted Gaussian basis set that includes for He 10s functions grouped in a [6,1,1,1,1] contraction and two sets of Cartesian p functions. Hydrogen is described with 6s/[3,1,1,1] and l p set functions. The basis functions for 0 are l l s / [ 5 , l , l , l , l , l , l ] , 7p/[3,1,1,1,1], and 2d/[l,l] Cartesian sets. For a test of this basis set, computations were also carried out with Langhoff et a1.I8 basis set I11 used in the study of OH. This basis set is 14s/ [6,2,1,1,1,1,1,1], 7p/[3,1,1,1,1], and 2 d / [ l , l ] for oxygen and 6s/[3,1,1,1] and l p for hydrogen. Diffused p and d orbitals suitable to describe the Rydberg states of OH were also included. According to the MRD-CI method the configuration space is generated by all single and double excitations relative to nine main configurations for A' states and eight for the A" states. So that the dimensions of the secular matrices could be kept in the range of 4000-5000, only configurations lowering the energy by more than a threshold of 20 Khartrees were included. The contribution of the remaining configurations is accounted for by an extrapolation scheme." The error in the fit is about dz3 mhartrees in the repulsive part of the potential and f l mhartree at the dissociation limits. The sum of squares of the coefficients of the main configurations was in the range 0.92-0.95. The S C F molecular orbitals are those of the A' state. However, calculations with SCF-MO of the A" state gave differences in the energy values of the 1A" state comparable to the extrapolation error. For each symmetry, A' and A", we compute two roots as a function of the distance of He from the center of mass of OH, R, the bond length of O H , r, and the angle between those distances, 29. 29 = Oo is defined when He approaches oxygen. So that the quality of the basis set could be tested, in Table I the energies of the first three surfaces are compared with those obtained by Langhoff et a1.I8 atomic functions 111. The smaller basis set that we have used systematically gives lower values. Table I1 shows the energies of OH(X211,A2Y)with H e at a distance of 10 a. and these are compared with previous calculations. We (28) Garrison, B. J.; Lester, W. A., Jr.; Schaefer, H. F., I11 J. Chem. Phys. 1975, 63, 304.
The Journal of Physical Chemistry, Vol. 92, No. 10, 1988 2725 TABLE 111: Comparison of Different Asvmototic Energies
system O('D) - o ( 3 ~ )
He('S) W2S) HeH(X22+)(at 3.0 ao) HeH(X2EC)(at 2.5 ao)
this work
Elhartree literature values
-74.993 097 0.076 01 1 -2.896 454 -0.499 187 -3.383 48 -3.366 39
-74.947 9" 0.072 O3Ob -2.896 473' -0.499 1" -3.383 O l d -3.366 65d
'MRD CI calculations from ref 23. bExperimental value.24 CFrom ref 25. dFrom ref 7. TABLE IV: Potential Energies (hartree) for the First Three States of H e 0 Rlan 'IIOBiI 'Z+('A,l 32-(3A,) 1.6 -77.709 64 1.7 -77.134 28 1.8 -77.55904 -77.752 33 -77.262 94 2.0 -77.771 57 -77.666 10 -77.448 63 2.1 -77.776 87 2.2 -77.73491 -77.781 05 -77.58406 2.3 -77.785 32 2.4 -77.787 89 -77.784 52 -77.681 81 2.5 -77.788 97 -17.802 25 -77.71764 2.7 -77.793 51 2.8 -77.799 94 -77.841 89 -77.794 73 -77.798 77 3.0 -77.851 94 -77.823 92 3.2 -77.86941 -77.805 05 -77.845 95 3.4 -77.804 97 -77.875 52 -77.862 68 3.5 -77.878 37 -77.867 15 3.6 -77.881 11 -77.807 21 -77.871 12 3.7 -77.807 64 3.8 -77.81145 4.0 -77.809 5 1 -77.885 95 -77.88200 4.4 -77.809 53 4.5 -77.888 46 -77.885 83 5.0 -77.889 12 -77.885 83 6.0 -77.889 55 -77.814 85
He+ O(3P.'D)
050 h
v
W
OWo_
L R (ao)
Figure 1. Potential curves for HeO.
underline the good agreement of the full C I values with the energies of Langhoff et a1.I8 obtained with the Slater type basis set I. Finally the dissociation limits to atomic fragments and HeH products are given in Table I11 and are compared with previous results. Agreement is obtained within the error of our computations.
3. Results H e 0 Potential Curues. There are no previous calculations in the literature for the repulsive potentials of HeO. We computed the first three potential curves of this system. These are the 311 and 32-states, which dissociate to ground-state atoms, and the 'E+,which dissociates to O('D) and He(%). The results are given in Table IV, and the potential curves are shown in Figure 1. In accord with Hund's rules we remark the following: (1) For distances greater than 2.8 a. both triplet states have lower energies than the singlet I F . (2) The 311 is the ground state. This is
2726 The Journal of Physical Chemistry, Vol. 92, No. 10, 1988
Vegiri and Farantos
TABLE V 1A' Potential Enerm Surface (hartree) of HeOH" R/ao r = 1.5 a. r = 1.6 a,, r = 1.7 a. r = 1.83 a. I9 = 450 I -
1.5 2.0 2.25 2.5 3.0 4.0
-0.470 53 -0.488 45 -0.499 74
-0.31767
-0.331 73
-0.495 71 -0.513 08 -0.525 43
-0.509 17 -0.526 34 -0.537 73 -0.54088
-0.334 76 -0.48 1 84 -0.5 14 32 -0.529 38 -0.542 56 -0.544 68 -0.331 -0.410 -0.464 -0.512 -0.539 -0.544
r = 2.5 a.
r = 3.0 a.
-0.426 19
-0.378 97
-0.47008 -0.485 66 -0.491 65
-0.427 95 -0.440 24 -0.441 14
00
-0.279 49 -0.35699 -0.41 184 -0.459 08 -0.485 42 -0.491 63
-0.249 68 -0.31033 -0.365 04 -0.417 48 -0.444 73 -0.444 95
-0.268 98 -0.452 05
-0.397 28
-0.349 98
-0.51627 -0.533 13 -0.537 61
-0.455 75 -0.476 83 -0.484 83
-0.41 5 06 -0.428 89 -0.440 51
r = 2.5 a.
r = 3.0 a.
-0.399 30
-0.356 58
-0.467 56 -0.486 14 -0.489 09
-0.423 84 -0.440 16 -0.435 71
r = 2.0 a. -0.328 -0.473 -0.506 -0.523 -0.535 -0.538
08 06 77 37 55 44
-0.319 -0.402 -0.456 -0.506 -0.534 -0.538
54 96 10 96 38
r = 3.5 a,
-0.386 12 -0.402 97 -0.41474
I9 = 90° 2.0 2.25 2.5 3.0 4.0 10.0 1.5 2.0 2.25 2.5 3.0 4.0
-0.37025 -0.421 56 -0.472 97
-0.394 75 -0.446 65 -0.498 45
-0.50303
-0.528 35
-0.407 74 -0.458 57 -0.511 81 -0.537 13 -0.54096
-0.249 56
-0.261 08
-0.491 54 -0.510 15 -0.525 00
-0.502 -0.522 -0.536 -0.541
-0.46945 -0.489 01 -0.500 46
93 37 88 36
47 58 79 76 75 59
I9 = 135O -0.268 67 -0.463 70 -0.503 03 -0.525 09 -0.541 27 -0.544 99
-0.284 66 -0.335 59 -0.388 18 -0.41622
-0.366 29 -0.383 02 -0.402 66
Energies are relative to -78.0 hartree. TABLE V I 1A" Potential Energy Surface (hartree) of HeOH' R/ao r = 1.5 a. r = 1.6 a. r = 1.7 a,, r = 1.83 a 1.5 2.0 2.25 2.5 3.0 4.0 2.25 2.5 3.0 4.0 10.0 1.5 2.0 2.25 2.5 3.0 4.0 a
,
r = 2.0 a. -0.180 91 -0.447 5 5 -0.494 07 -0.5 16 74 -0.533 50 -0.538 40 -0.273 69 -0.373 54 -0.475 82 -0.53005 -0.538 00
-0.322 -0.426 -0.479 -0.488
-0.153 84 -0.426 59
-0.372 09
-0.326 41
-0.51059 -0.53099 -0.538 68
-0.451 38 -0.478 55 -0.485 56
-0.404 85 -0.432 64 -0.444 29
-0.164 17
-0.17829
-0.458 22 -0.481 54 -0.496 78
-0.482 54 -0.506 00 -0.52290
-0.495 -0.518 -0.533 -0.540
25 88 48 44
I9 = 450 -0.18321 -0.452 89 -0.500 98 -0.523 50 -0.540 18 -0.544 66
-0.238 2 -0.337 56 -0.439 58
-0.262 55 -0.362 24 -0.464 21
-0.502 63
-0.527 63
-0.275 -0.375 -0.477 -0.531 -0.540
52 69 21 65 17
I9 = 90° -0.28031 -0.38004 -0.482 88 -0.536 21 -0.545 09
-0.1 36 94
-0.147 38
-0.477 86 -0.502 97 -0.522 84
-0.488 71 -0.51488 -0.533 92 -0.539 57
-0.455 01 -0.480 95 -0.497 24
I9 = 135" -0.15362 -0.437 49 -0.490 35 -0.51835 -0.539 62 -0.545 33
r = 3.5 a.
-0.371 05 -0.392 90 -0.41 145
-0.207 35 -0.28 1 40 -0.38047 -0.438 99
71 04 79 43
-0.224 47 -0.283 11 -0.355 85 -0.41641
-0.35223 -0.375 18 -0.399 19
Energies are relative to -78.0 hartree.
understood since in the leading configuration of the wave functions we find one and two electrons in the pz orbital, which is across the molecular axis, for the 311 and 32-states, respectively. On the other hand, in the singlet state there are no electrons in the pz orbital, which allows H e to approach 0 close enough before experiencing the repulsive forces. (3) This leads to a potential curve that is flatter than triplets, and that results in a curve crossing between 32-and '2+at a distance of 2.8 a. and 0.4 eV above the dissociation limit O('D) + He(%). Similarly, the I P curve crosses the 311at a distance of 2.5 a. and 0.9 eV above the dissociation limit. It is worth comparing these values with the corresponding ones of NeO, ArO, KrO, and XeO of Dunning and Hay.27 The helium oxide shows the smallest crossing distance, verifying the trend that the distance at the crossing points increases in going from H e 0 to XeO (see Table 9 of ref 27). However the energy differences, l2+ - 311 and lZ+ - 32-,at the crossing points for H e 0 are comparable to N e 0 without showing the differences observed between N e 0 and ArO. HeOH Potentials. All interactions of He + O H are found to be repulsive. W e did not try to locate van der Waals minima, as their depths will be comparable to the error of these computations. Calculations were performed for the O H bond lengths
L
01
l26,,
la%
00
1
2
0
RlaJ
4
S
B
Z
J
4
5
8
Wa,)
Figure 2. Potential curves for HeOH; I9 = Oo when He approaches 0.
at 1.5, 1.6, 1.7, 1.83, 2.0, 2.5, 3.0, and 3.5 ao. The distance of He from the center of mass of O H was equal to 1.5, 2.0, 2.25, 2.5, 3.0, 4.0, 6.0, and 10 a. and for the angles 19 equal to,'O 45", 90°, 135', and 180'. About 150 energies were calculated altogether and are presented in Tables V-VI1 for C, geometries, and the results for collinear configurations are shown in Table VIII. For the 12A' state the configuration with the highest coefficient la'2a' MOs are mainly is the (la')2(2a')2(3a')2(4a')z(la'')2(5a')1. the 1s and 2s oxygen orbitals, whereas the 3a' is the 1s orbital
He(%)
+ OH(X211) and He(%) + OH(A2Z+)
The Journal of Physical Chemistry, Vol. 92, No. 10, 1988 2727
TABLE VII: 2A‘ Potential Energy Surface (hartree) of HeOH“ Rian r = 1.5 a, r = 1.6 a, r = 1.7 a , r = 1.83 no I9 = 450 -0.097 37 -0.104 13 1.5 -0.081 99 -0.329 28 2.0 -0.347 98 -0.361 20 2.25 -0.307 16 -0.334 53 -0.375 28 -0.363 73 2.5 -0.322 10 -0.349 79 -0.374 96 -0.387 66 3.0 -0.333 23 -0.362 14 -0.380 59 -0.390 47 4.0 2.0 2.25 2.5 3.0 4.0 10.0 1.5 2.0 2.25 2.5 3.0 4.0
-0.08068 -0.17741 -0.275 34
-0.10633 -0.203 68 -0.303 99
-0.337 46
-0.36488
-0.297 65 -0.319 50 -0.33361
-0.12395 -0.218 53 -0.31871 -0.371 76
-0.01965
-0.037 92
-0.323 98 -0.344 57 -0.361 01
-0.337 92 -0.361 08 -0.376 25 -0.381 08
I9 = 90° -0.148 9 1 -0.144 14 -0.227 62 -0.326 89 -0.378 61 -0.390 64 19 = 135O -0.053 39 -0.307 83 -0.349 44 -0.371 19 -0.385 31 -0.39008
r = 2.0 a.
r = 2.5 a,
r = 3.0 a.
-0.288 77
-0.262 70
-0.339 40 -0.353 13 -0.35621
-0.312 17 -0.320 18 -0.323 40
-0.216 -0.224 -0.227 -0.294 -0.346 -0.355
17 84 58 01 20 58
-0.210 74 -0.251 56 -0.267 21 -0.28068 -0.32074 -0.324 19
-0.064 82 -0.303 94
-0.274 51
-0.243 89
-0.369 93 -0.383 95 -0.389 53
-0.332 40 -0.350 33 -0.355 51
-0.304 56 -0.316 13 -0.321 74
-0.10944 -0.324 78 -0.359 44 -0.375 37 -0.386 54
-0.16244 -0.226 04 -0.326 81 -0.380 55
r = 3.5 a,,
-0.286 14 -0.298 78 -0.313 29
-0.266 93 -0.275 49 -0.282 89 -0.3 12 35
-0.266 63 -0.290 9 1 -0.302 09
“Energies are relative to -78.0 hartree.
TABLE VIII: Potential Energies (hartree) for Collinear Geometries” R / a n r = 1.83 a,, r = 2.0 an r = 2.5 a,, r = 3.0 an State 12B,,2: I9 = Oo -0.094 04 -0.188 14 -0.1 16 52 2.0 -0.381 27 -0.377 97 2.5 -0.391 09 -0.486 68 -0.481 40 -0.434 84 3.0 -0.538 73 -0.532 61 -0.485 91 -0.438 33 4.0 -0.545 46 -0.539 52 -0.49068 5.0 -0.54571 -0.53969 -0.49081 6.0
I9 = 180° 3.0 3.5 3.7 4.0 4.5 5.0 6.0
-0.374 38
-0.2 17 9 2 -0.330 83
-0.523 88
-0.51 193
-0.43247
-0.542 98 -0.544 72
-0.534 42 -0.537 96
-0.477 83 -0.486 98
2.0 2.5 3.0 4.0 5.0 6.0
-0.240 15 -0.338 16 -0.36706 -0.389 56 -0.391 23 -0.391 38
3.0 3.5 4.0 4.5 5.0 6.0
-0.295 79
State 12AI: I9 = 0’ -0.232 -0.337 37 -0.367 53 -0.335 -0.388 18 -0.356 -0.389 48 -0.356 -0.39055 -0.356
-0.148 17 -0.143 83 -0.388 45 -0.419 38 -0.436 77
95
-0.232 59
68 53 79 16
-0.3 13 00 -0.325 18
-0.38394 -0.39044 -0.39062
-0.381 87 -0.388 1 1 -0.38995
-0.354 13 -0.357 18
A procedure used by other investigator^^^,^^ can be followed to obtain the two components of the II state. This involves the fitting of the sum and difference of the three-body interactions: a+ = Y2(vAJ3) vAJ3)) (12)
+
a- = f/2(VA,(3)- VA,,(3)) VAA3)= @+
-0.10789 -0.303 67 -0.315 15 -0.32299
“I9 = 0’ when H e approaches 0. Energies are relative to -78.0 hartree.
+ a-
- @a* are
VJ3) = @+ The functional forms chosen for 4
@+ =
C(all+ a12p+ a 1 3 p 2 ) e - Y ~ + R P8) I ( ~ ~ ~ (14)
/=0
a- = 2 ( b l l + b12p+ b13p2)e-yl-RP;(~os8 ) /=2
of helium. Similarly, for the 12A” and 22A’ states the leading configurations are (1a’)2(2a’)2( 3a’)2(4a’)2( 1a”)l and ( la’)2(2a’)2(3a’)2( 1a”)2(4a’)1(5a’)2, respectively. If x-z is the molecular plane, then we expect the 12A’ state t o be of lower energy than the 12A” since the la” MO is mainly the px orbital of oxygen. In Figure 2 we show potential curves for the collinear approach of H e to OH at r = 1.83 ao. The 22A’ state is flatter than the ground state for the same reasons given in H e 0 case. This results in a crossing of these states at R = 2.3 a. and 2.0 eV above the dissociation limit He(IS) OH(A2Z+). Analytical Potential Functions. A general method of modeling the Renner-Teller intersections of II symmetry states and a conical
+
(13)
From these we get
I9 = 1800
-0.236 06 -0.274 14 -0.341 00
intersection between I; and II states has been proposed by Murrell the diagonalization of a 3 X 3 matrix and C a ~ t e r . ~This ~ . involves ~~ which is factored into a 2 X 2 block of A‘ symmetry and a 1 X 1 block of A” symmetry. The interaction term of the A’ block is chosen such as to become zero for linear configurations, and the A’’ and one of the two A’ diagonal elements are chosen to become equal at linear geometries, while the remaining A’ diagonal element represents the I; state. Since the intersection of the 211 and 2Z states of HeOH occurs at an energy rather high for the present experimental interests, single-valued analytical functions, which fit those regions of the pes that have been studied in these computations, seems appropriate. Such functions used in dynamical calculations will be more economical in computer time. The potentials can be written as a sum of the O H diatomic potential and the interaction of H e with OH: V(r,R,8) = VoH(r)+ f13)(r,R,3) (11)
(15)
p=r-rc P/(cos 9 ) and P;(cos 8) are Legendre and associated Legendre polynomials, respectively. The P: functions guarantee the correct (29) Herzberg, G. Molecular Spectra and Molecular Structure; Van Nostrand Reinhold: New York, 1950. (30) Murrell, J. N.; Carter, S.;Mills, I. M.; Guest, M. A. Mol. Phys. 1981, 42, 605. Knowles, P.; Handy, N. C.; Carter, S. Mol. Phys. 1983; 49, 681. Carter, S.;Dixon, R. N. Mol. Phys. 1985, 55, 701. (31) Murrell, J. N.; Carter, S.; Farantos, S.C.; Huxley, P.; Varandas, A. J. C. Molecular Potential Energy Functions; Wiley: New York, 1984. (32) Rebentrost, F.; Lester, W. A., Jr. J. Chem. Phys. 1975, 63, 3737. Ibid. 1976, 64, 3879. (33) Alexander, M. H.; Corey, G. C. J . Chem. Phys. 1986, 84, 100.
2728
The Journal of Physical Chemistry, Vol. 92, No. 10, 1988
TABLE IX: Three-Body Term of the Excited Potential Function of He(%) OH(A*Z+)"
+
PO 519.5865 -64.6542 ( ~ 3 l -493.4210 4.05 =
PI -187.9795 -43.6897 152.0297 4.10
x$,,(a,l,+
P, -256.7397 -2.2649 1434.7350 4.05
P1 -630.8080 0.0 0.0 4.10
PA
751.7195 -209.5946 -338.591 1 4.00
a 2 p + a31p2)e-7'RP,(~os 8). p = r - 1.0107.
Energy in eV and distances in
A.
3 00,
I
I
I
c
1
1 4
L
c -3 0 0 ' -3 00
I 0 00
x /A
3 00
Figure 3. Potential contours for He + OH(A2Z+).The energies of the plottedcontoursare0.05,0.10,0.15,0.20,0.30,0.40,0.50, 0.60,0.80, and 1.0eV.
topological behavior of the Renner-Teller intersection at 8 = OD and 180°.32 The coefficients in eq 14 and 15 are determined by a least-squares fit to the a b initio calculations:
v(3)(r,,R,,8i)= E(r,,Ri,8,)- E ( r i , R = - ) (16) Here we present an analytical fit for the excited He + OH(?Z+) surface. The three-body term has been fitted to a form given by eq 14, and the coefficients are tabulated in Table IX. The equipotential contour plots of this function are shown in Figure 3 in Cartesian coordinates. 4. Discussion The main topological characteristics of the three potential energy surfaces are the following: (1) The minimum energy
Vegiri and Farantos approach of helium to hydroxyl is across the angles 45O and 135'. (2) By subtraction of the two-body term from the total energy, eq 16, it is shown that there is no strong coupling between r a n d R coordinates. That will lead to an inefficient vibrational excitation or deexcitation of O H in collisions with He. Apparently this has been observed by Lengel and C r o ~ l e y .They ~ estimate a cross section of 0.09 f 0.03 At for the vibrational deexcitation of O H ( A Z P , ~1)= in room-temperature collisions. (3) The electronic quenching of the A2Z+ state is not expected to be an open channel for collision energies lower than 2.0 eV. At this energy and for collinear geometries the excited pes crosses the ground 211state. (4) Inspection of Table VI1 reveals that the 2*A' pes is leveling off at 8 = 90' and R = 2.25 ao. Calculations of the third root of A' symmetry at these geometries show that there is an avoided crossing of this state with the 22A'. This topological characteristic occurs at high energies, and therefore we do not study it any further. ( 5 ) Figure 3 shows the high anisotropy of the potential function. This is expected to lead to efficient rotational energy transfer. As far as the ground state of O H is concerned, preliminary experimental work by Andresen and cow o r k e r ~with ~ ~ crossed beam experiments show large ratios of the cross sections for the two components of the rotational angular momentum. However, dynamical calculations are necessary to resolve question related to the A-doublet population. The present calculations have not been designed to evaluate the van der Waals minima. Ab initio quantum chemistry calculations on the ground state of He0,26aHeOz,26band HeHzZs showed that the depths of the van der Waals minima are 1-3 meV. Such shallow minima require a very small error in the computations. From these results it is reasonable to expect an analogous van der Waals minimum for the excited 22A' surface of HeOH. Thus for high-energy collisions such a minimum will play a minor roll. Therefore, we consider the constructed potential energy surface for H e OH(A28+) adequate for studying collisions in the range of 50-200 meV, the results of which are presented in the companion paper.
+
Acknowledgment. We thank Prof. Buenker for making the MRD-CI programs available to us. We are also grateful to Prof. Alexander for stimulating discussions. OH, 3352-57-6. Registry No. He, 7440-59-7;
(34)Andresen, P.,private communication
