J . Phys. Chem. 1985, 89, 5336-5343
5336
not change the barrier heights.22 We have determined the barrier heights at the CI-SD level with the optimum S C F .geometries for consistency. This should not introduce more than 0.1-0.2 kcal/mol of error in the barrier heights. In order to compare with an experimental value, zero-point corrections to the barrier height must be included since the cis and trans structures have one less active mode than the skew structure. The zero-point energies (ZPE) are given in Table 111. To a good approximation, the difference in zero-point energies is due to the loss of the torsional mode. Subtraction of this mode from the ZPE for the optimum structure gives 11.8 1 kcal/mol, in good agreement with the ZPE's for the cis and trans structures. In order to account for the effects of correlation and anharmonicity on the ZPE, we have scaled the difference in ZPE's by the ratio ZPE,,,,I/ZPE,Id for the skew structure which is 0.900. The final barriers including the scaled barrier heights are 5.02 and 7.54 kcal/mol for trans and cis, respectively. We suggest that a good estimate for the trans barrier is 5.0 & 0.15 kcal/mol and for the cis barrier it is 7.5 f 0.15 kcal/mol. Redington2' analyzed the torsional band in the far-infrared (22) The effeot of higher-order excitations can be included with the use of Davidson's formula for unlinked clusters (Langhoff, S. R.; Davidson, E. R. Int. J . Quantum Chem. 1974,8, 61). These effects are again small, -0.1 kcal/mol, lowering the trans barrier to 5.55 kcal/mol and the cis barrier to 8.08 kcal/mol (E(C1-SDQ, SCF opt) = -796.472 548 kcal/mol). Since the formula is only approximate, we are probably overestimating the effect of higher-order excitations since the size of the correction is the same as the CI-SD correction.
spectrum. Using a model potential, he estimated a barrier of 6.9 kcal/mol. This barrier could be considered as the average of the cis and trans barrier. Our average value is 6.25 kcal/mol which is slightly lower. This agreement is quite good considering the approximate form of the potential used in interpreting the experimental results and the fact that the experimental result is extrapolated from the low-lying modes. We have examined the dependence of the dipole moment, p, and the ionization potential, IP, on T . For the optimum structure, the value of p is 1.44 D at the CI-SD level (p(SCF) = 1.51 D). The S-H bonds are aligned at T = 0' giving p(C1-SD) = 2.02 D (p(SCF) = 2.12 D) while at T = 180°, there is no dipole moment since the S-H bonds are aligned in opposite directions. The ionization potential from Koopmanns' theorem is predicted to be 10.43 eV at the optimum value of T with the NHOMO (next highest occupied molecular orbital) slightly more stable at 10.52 eV. At T = O', these orbitals split apart with the IP decreasing to 9.17 eV and the N H O M O stabilized at 12.44 eV. A similar splitting is found at T = 180' with the IP = 9.13 eV and the NHOMO at 12.32 eV. In conclusion, we have determined the structure for H2S2and found it to be in good agreement except for the value for B(HSS). We suggest that the experimental value is too small. We have also determined the cis and trans rotation barriers and have shown that the correction for zero-point energy effects is not negligible. The barriers to internal notation are of a reasonable enough size that splitting of the higher vibrational levels of the torsional band could be observed experimentally.
FEATURE ARTICLE The F -t H, Potential Energy Surface: The Ecstasy and the Agony Henry F. Schaefer I11 Department of Chemistry and Lawrence Berkeley Laboratory, University of California, Berkeley, California 94720 (Received: June 13, 1985)
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This account surveys 14 years of more or less continuing theoretical research on the FH2potential energy hypersurface. Early encouragement concerning the ability of theory to reliably characterize the entrance barrier for F + H2 FH + H has more recently been sobered by the realization that very high levels of theory are required for this task. The importance of zero-point vibrational corrections and tunneling corrections in reliable predictions of the same activation energy is discussed. In contrast, the barrier height of H + FH HF + H three-center exchange stands as a prominent early success of ab initio molecular electronic structure theory
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Introduction It is possible to make a strong case that the elementary molecular process F + H2 ---* F H + H (1) is now the best understood of all chemical reactions.',* It may be stated with confidence that the only serious competitor is the simpler reaction3 H H,+H2 + H (2)
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(1) D. E. Neumark, A. M. Wcdtke, G. N. Robinson, C. C. Hayden, and Y . T. Lee, Phys. Reu. Left.,53,226 (1984); J. Chem. Phys., 82,3045 (1985). (2) D. M. Neumark, A. M. Wcdtke, G. N. Robinson, C. C. Hayden, K. Shobatake, R. K. Sparks, T. P. Schafer, and Y . T. Lee, J . Chem. Phys., 82, 3067 (1985).
0022-3654/85/2089-5336$01 .SO10
The F + H2 reaction (1) has been studied many times over the past 15 years by using the traditional methods of by (3) See, for example, D. P. Gerrity and J. J. Valentini, J . Chem. Phys., 82, 1323 (1985). (4) K. Homann, W. C. Soloman, J. Warnatz, H. G. Wagner, and C. Zetzsch, Ber. Bunsenges. Phys. Chem., 74, 585 (1970). ( 5 ) A. F. Dodonov, G . K. Lavrovskaya, I. I. Morozov, and V. L. Tal'roze, Dokl. Akad. Nauk SSSR, 198, 622 (1971) (p 440 in English edition). (6) R. Foon and G. R. Reid, Trans. Faraday SOC.,67, 3573 (1971). (7) R. L. Williams and F. S. Rowland, J . Phys. Chem., 75,2709 (1971); 77, 301 (1973). (8) K. H. Homann and D. I. MacLean, Ber. Bunsenges. Phys. Chem., 75, 945 (1971). (9) R. Foon, G. P. Reid, and K. B. Tait, Trans. Faraday SOC.,68, 113 1 (1972). (10) K. L. Kompa and J. Wanner, Chem. Phys. Lett., 12, 560 (1972).
0 1985 American Chemical Society
The Journal of Physical Chemistry, Vol. 89, No. 25, 1985 5337
Feature Article which is meant methods designed to determine overall rate constants and activation energies. In addition, newer techniques have been developed which allow the measurement of relative rates involving the different vibrational states of the product H F m ~ l e c u l e . ~ " ~Finally, there is growing e ~ i d e n c e ' ,that ~ - ~the ~ F H2 reaction is the first for which quantum-mechanical resonances have been observed experimentally. In terms of understanding the potential energy surface governing an elementary chemical reaction, the most important parameter extracted from traditional kinetic studies is the Arrhenius activation energy, E,, defined by the expression
Relative Cross Sections for F+H, 800 1
+
k(7') = A exp(-E,/RT)
(3)
for the rate constant k( 7'). For many years the 1959 value E, = 1.7 1 kcal/mol deduced by Mercer and PritchardU [who actually measured k(F + H z ) / k ( F + CH,)] was generally accepted for the F Hz reaction. Anderson notes in his 1980 review45that
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(11) S. W. Rabideau, H. G. Hecht, and W. B. Lewis, J . Magn. Reson., 6, 384 (1972). (12) R. K. Pearson, J. 0.Cowles, G. L. Herman, D. W. Gregg, and J. R. Creighton, IEEE J. Quantum Electron., 9, 879 (1973). (13) A Persky, J . Chem. Phys., 59, 5578 (1973). (14) M. A. A. Clyne, D. J. McKenney, and R. F. Walker, Can. J . Chem., 51, 3596 (1973). (15) T. M. Lam, M. Peyron, and P. Puget, J. Chim. Phys. Phys.-Chim. Biol., 71, 377 (1974). (16) E. R. Grant and J. W. Root, Chem. Phys. Lett., 27,484 (1974); J. Chem. Phys., 63, 2970 (1975). (17) F. S. Klein and A. Persky, J . Chem. Phys., 61, 2472 (1974). (18) R. Foon and M. Kaufman, Prog. React. Kinet., 8, 81 (1975). (19) R. G. Manning, E. R. Grant, J. C. Merrill, N. J. Parks, and J. W. Root, Inr. J . Chem. Kinet. 7 , 39 (1975). (20) I. B. Goldberg and G. R. Schneider, J . Chem. Phys., 65, 147 (1976). (21) W. E. Jones and E. G. Skolnik, Chem. Rev., 76, 563 (1976). (22) N. Cohen and J. F. Bott in "Handbook of Chemical Lasers", R. W. F. Gross and J. F. Bott, Eds., Wiley, New York, 1976. (23) C. R. Quick and C. Wittig, Chem. Phys. Left., 40, 420 (1977). (24) J. M. Preses, R. E. Weston, and G. W. Flynn, Chem. Phys. Lett., 48, 425 (1977). (25) D. J. Smith, D. W. Setser, K. C. Kim, and D. J. Bogan, J . Phys. Chem., 81, 898 (1977). (26) R. F. Heidner, J. F. Bott, C. E. Gardner, and J. E. Melzer, J . Chem. Phys., 70, 4509 (1979). (27) E. Wurzberg and P. L. Houston, J . Chem. Phys., 72,481 1 (1980). (28) R. F. Heidner, J. F. Bott, C. E. Gardner, and. J. E. Melzer, J . Chem. Phys., 72, 4815 (1980). (29) M. A. A. Clyne and A. Hodgson, J . Chem. SOC.,Faraday Trans. 2, 81, 443 (1985). (30) J. H. Parker and G. C. Pimentel, J . Chem. Phys., 51, 91 (1969). (31) J. C. Polanyi and D. C. Tardy, J . Chem. Phys., 51, 5717 (1969). (32) T. P. Schafer, P. E. Siska, J. M. Parson, F. P. Tully, Y. C. Wong, and Y. T. Lee, J . Chem. Phys., 53, 3385 (1970). (33) K. G. Anlauf, P. E. Charters, D. S. Horne, R. G. Macdonald, D. H. Maylotte, J. C. Polanyi, W. J. Skrlac, D. C. Tardy, and K. B. Woodall, J . Chem. Phys., 53, 4091 (1970). (34) N. Jonathan, C. M. Melliar-Smith, S. Okuda, D. H. Slater, and D. Timlan, Mol. Phys., 22, 561 (1971). (35) J. C. Polanyi and K. B. Woodall, J . Chem. Phys., 57, 1574 (1972). (36) H. W. Chang and D. S. Setser, J . Chem. Phys., 58, 2298 (1973). (37) R. D. Coombe and G. C. Pimentel, J . Chem. Phys., 59, 251, 1537 (1973). (38) M. J. Berry, J . Chem. Phys., 59, 6229 (1973). (39) D. S. Perry and J. C. Polanyi, Chem. Phys., 12, 37, 419 (1976). (40) D. J. Douglas and J. C. Polanyi, Chem. Phys., 16, 1 (1976). (41) J. C. Polanyi and J. L. Schreiber, Faraday Discuss. Chem. SOC.,62, 267 (1977). (42) 0. D. Krogh, D. K. Stone, and G. C. Pimentel, J . Chem. Phys., 66, 368 (1977). (43) R. K. Sparks, C. C. Hayden, K. Shobatake, D. M. Neumark, and Y. T. Lee, "Horizons of Quantum Chemistry", K. Fukui and B. Pullman, Eds., Reidel, Dordrecht, Holland, 1980, pp 91-105. (44) P. D. Mercer and H. 0. Pritchard, J . Phys. Chem., 63, 1468 (1959).
5
I 4v)
1
, 400 0 ° '
/ 2
00 A--
b 200 -
0 , 15
........3......0 .......................... ........................... ...................
I
2
0
- -J, ---- - +, 25
35
3
collision energy(kcal/mole) Figure 1. Relative cross sections for F + H 2 as a function of collision energy from the molecular beam studies of Neumark, Wodtke, Robinson, Hayden, and Lee.' The curves labeled 1, 2, and 3 refer to cross sections for the product HF molecule in vibrational states u = 1, 2, and 3, respectively. The top (unlabeled) curve is the sum of the cross sections for u = 1, 2, and 3.
"the first direct measurements leading to an activation energy" were those of Homann, Soloman, Warnatz, Wagner, and Zetzsch? who found E, = 1.6 kcal/mol. Reviewing all the experimental data as of 1979, Anderson concluded, "It seems highly unlikely that the activation energy could lie outside the range 1.3 to 1.9 kcal/mol". The two most recent experimental values of the F H2 activation energy are significantly lower than the earlier values. Wurzberg and Houston2' in 1980 reported E, = 0.86 f 0.10 kcal/mol. Simultaneously Heidner, Bott, Gardner, and MelzerZ8 concluded that E, = 1.18 f 0.10 kcal/mol. Thus, the two experiments are mildly incompatible, their ranges being 0.76-0.96 and 1.08-1.28 kcal/mol. The first experiments presenting information concerning relative rates into the different product vibrational levels were the chemical laser studies of Parker and PimenteL30 In their 1969 paper, Parker and Pimentel reported k(u=2,HF) 5.5 (4) k(u=l,HF)
+
-
Later the same year Polanyi and Tardy3' used the method of infrared chemiluminescence to determine ratio (4)to be 13.5. Polanyi and Tardy also found k(u=3,HF) R 0.47 (5) k(u=2,HF) Refinements in both the chemical laser and infrared chemiluminescence techniques in subsequent years led to much more precise product vibrational rate ratios. The currently accepted ratios at 300 K are45 (u = 3):(u = 2):(u = l):(u = 0) = 12:20:6:1 There is agreement all around that the u = 2 vibrational level of H F is most highly populated, followed by u = 3 and u = 1, with relatively little u = 0 H F populated. The first crossed molecular beam experiment to give vibrationally resolved product results was that of Schafer, Siska, Parson, Tully, Wong, and Lee32reported in 1970. Despite continuing research on this problem, Lee and co-workers waited 10 years to present a second preliminary report43on the F H2reaction. And they waited another 5 years to publish their full paper.' But what a paper it is! Their vibrationally state-resolved differential cross sections for F + H2 certainly represent the most in-depth experimental study in reaction dynamics to date. Figure 1 shows
+
(45) J. B. Anderson, Adu. Chem. Phys., 41, 229 (1980).
5338 The Journal of Physical Chemistry, Vol. 89, No. 25, I985 Lee and co-workers relative cross sections as a function of collision energy. These results confirm the less precisely specified chemical laser and IR chemiluminescence results of Pimentel and Polanyi's groups, finding u = 2 > u = 3 > u = 1. The experiments of Neumark, Wodtke, Robinson, Hayden, and Lee1 strongly suggest that dynamical resonances46play a significant role in the reaction dynamics of F + H2. The purpose of this account is to review theoretical progress over the past 14 years on the ab initio understanding of the F + H 2 potential energy surface. During this period computational quantum chemistry has had many succe~ses,4~*~ but one must state at the outset that the FH2 story has in some respects been discouraging. Nevertheless, a complete assessment of the state of predictive molecular quantum mechanics in 1985 must include the F H2 potential energy surface.
+
Schaefer TABLE I: Constrained Saddle Point Geometries and Energies for Different Angles '6 of ApproachQ
0, den 0 10 30 50 70 90
R(F-H), bohrs
R(H-H), bohrs
energy, kcal/mol
2.58 2.55 2.50
1.54 1.54 1.56 1.60 1.68 1.87
0.00 0.02 0.36 1.59 4.7 1 11.80
2.44
2.39 2.35
'From the 1972 research of BPOS.S4 TABLE 11: The "Ecstasy" of 1972, the BOPS Results6*for the F H2Potential Energy Surfaceo
property
SCF
CI
+
expt
A Reconnaissance Mission barrier height, kcal/mol 29.3 1.66 1.7 exothermicity, kcal/mol 13.2 34.4 31.8 Our interest in the F H2 potential energy surface began in saddle point geometry, A 1971, in the wake of the exciting new experimental studies F-H 1.18 1.54 (discussed above) in the research groups of Pimentel,,O P ~ l a n y i , ~ ~ H-H 0.836 0 767 and Lee.32 It was apparent that a theoretical study of the F "The most reliable activation energy in 1972 was thought to be the H2 surface was very much in order. However, the only systems 1.71 kcal/mol value of Mercer and P r i t ~ h a r d . ~ ~ for which reliable ab initio potential energy surfaces were available were H, (reaction 2)49and H4. Furthermore, concerning the latter the Hartree-Fock level of theory to explicitly consider the effects surface, shock tube experiment^,^^*^^ pertaining to the reaction of electron correlation.56 This allows a description of the inH2 + D2 2HD stantaneous (as well as averaged) repulsions between electrons (6) by way of Coulomb's law. cast serious on the reliability of the theoretical predicThe method of configuration interaction (CI)5' was chosen by tion~.~' BPOS to approach the correlation problem. Specifically, they The first ab initio study54of the F H2problem was designed used "first-order wave function^",^^ designed to incorporate the to be analogous to the Rubinstein-Shavitt treatments3 of the H4 structure-sensitive part of the electron correlation energy. In potential energy hypersurface. That is, a double-{ (DZ) basis general, such first-order wave functions include all orbital occuset5swas adopted, including two Is, two 2s, and two 2p,, 2py, and pancies in which at most one electron is assigned to an orbital 2p, functions centered on fluorine. Such pairs of basis functions beyond the valence sheLS9 It may be seen that such a wave (e.g., 1s and 1s' on F) may be thought of as describing the inner function will attempt to describe that part of the correlation energy and outer parts of the respective atomic orbitals. The resulting due to low-lying valence orbitals not occupied in the SCF wave flexibility within the molecular environment is of critical imfunction (7) or (8). For the F + H 2 system these orbitals may portance. Similarly, each hydrogen atom 1s atomic orbital was be thought of as the sixth (unoccupied) F atom 2p spin orbital described by two 1s basis functions. The entire D Z basis set may and the la, antibonding orbital of H2. For general geometry [C, thus be labeled F(4s2p), H(2s). point group (8)] and the DZ basis set described, there were 214 The Hartree-Fock or self-consistent-field (SCF) ground-state configurations in the first-order CI wave functions of BPOS.54 electron configuration for linear F + H 2 geometries is One of the more important findings of the BPOS study was that the transition state for F H2 FH + H is indeed collinear, 1U22u23u24u 184 2z+ (7) consistent with semiempirical models such as the London-Eyring-Polanyi-Sat0 method.60 However, BPOS found that potential while for general (nonlinear) geometries it is surface to be rather flat with respect to bending. For an F-H-H angle 10' away from collinear, the energy lies only 0.02 kcal/mol above that of the transition state. These results are summarized in Table I, which shows that, for even a rather extreme F-H-H As a first step in their theoretical procedure, Bender, Pearson, angle of 90°, the constrained saddle point lies only 11.8 kcal above ONeil, and this author (BPOS)s4 determined SCF wave functions that of the linear transition state. for each geometrical agreement. However BPOS went beyond The most important result of this preliminary study of the F H 2 surface was the large effect of electron correlation on the predicted barrier height. Specifically BPOS found that CI reduced (46) For theoretical discussions of quantum-mechanical resonances in F the predicted S C F barrier by 28.6 kcal, an enormous amount + H2 FH + H, see G . C. Schatz, J. M. Bowman, and A. Kupperman, J . Chem. Phys., 58,4023 (1973); S.L. Latham, J. F. McNutt, R. E. Wyatt, and compared to the experimental activation energy of Mercer and M. J. Redmon, J . Chem. Phys., 69, 3746 (1978); M. J. Redmon and R. E. P r i t ~ h a r d only , ~ ~ 1.71 kcal. Moreover, the predicted barrier of Wyatt, Chem. Phys. Lett., 63,209 (1979); E. F. Hayes and R. B. Walker in 5.72 kcal, although greatly improved over the SCF result. was 'Resonances", American Chemical Society, Washington, D.C., 1984, ACS still in poor agreement with experiment. Symp. Ser. No. 263, pp 493-513. (47) H. F. Schaefer, "Quantum Chemistry. The Development of Ab Initio The primary conclusions of BPOS may be taken verbatim from Methods in Molecular Electronic Structure Theory", Clarendon Press, Oxford, their paper: "This study shows clearly that the energetic aspects 1984. of a chemical reaction are the most difficult to describe ab initio. (48) H. F. Schaefer, Science, in press.
+
+
-
+
+
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(49) I. Shavitt, R. M. Stevens, F. L. Minn, and M. Karplus, J . Chem. Phys., 48, 2700 (1968). (50) S. H. Bauer and E. Ossa, J . Chem. Phys., 45, 434 (1966). (51) A. Burcat and A. Lifshitz, J . Chem. Phys., 47, 3079 (1967). (52) Note that this dispute was eventually resolved in favor of theory: A. Lifshitz, M. Bidani, and H. F. Carroll, J . Chem. Phys., 79, 2742 (1983). (53) M. Rubinstein and I. Shavitt, J . Chem. Phys., 51, 2014 (1969). (54) C. F. Bender, P. K. Pearson, S. V. O'Neil, and H. F. Schaefer, J . Chem. Phys., 56, 4626 (1972). (55) S. Huzinaga, J . Chem. Phys., . 42, 1293 (1965); T. H. Dunning, J . Chem. Phys., 53, 2723 (1970).
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(56) H. F. Schaefer, "The Electronic Structure of Atoms and Molecules: A Survey of Rigorous Quantum Mechanical Results", Addison-Wesley, Reading, MA, 1972. (57) I. Shavitt, "Modern Theoretical Chemistry", Vol. 3, H. F. Schaefer. Ed., Plenum Press, New York, 1977, pp 189-275. (58) H. F. Schaefer, J . Chem. Phys.. 55, 176 (1971). (59) H. F. Schaefer and F. E. Harris, Phys. Rec. Lett., 21, 1561 (1968). (60) See, for example, P. J. Kuntz, E. M . Nemeth, J. C. Polanyi, S . D. Rosner, and C E Yoing, J Chem P h k s . 44. 1168 (1966)
Feature Article
The Journal of Physical Chemistry. Vol. 89, No. 25, 1985 5339
1
HtHF
F w 2.
A view from the entrance channel of the 1972 BOPS potential
energy surface6' for collinear FH,. The Hartree-Fock approximation does not appear to yield reasonable results in this regard, since the correlation energies for the reactants, transition state, and products may be quite different. In addition, an extended basis set including polarization functions is needed to reliably predict barrier heights and exothermicities".
TbeEestasy: 1972 Having explored some of the terrain of the F + H, surface and established the difficulty of making reliable energetic predictions, Bender, O'Neil, Pearson, and this author (BOPS)61set out later the same year to do the job properly for collinear F + H,. Their work appeared in the June 30 issue of Science. Energies were determined for 150 different collinear geometries, and the resulting analysis showed good agreement with experiment. This was the first example larger than H, of an ah initio reactive potential energy surface in harmony with the known experimental facts. BOPS began by adding polarization functions to their earlier double-{or DZ hasis set. Specifically, a set of 3d functions was added to the F atom basis set, and a set of 2p functions (2p, 2pr 2pJ to each hydrogen atom. Their double-{plus polarization (DZ + P) basis set may be labeled F(4s2pld), H(2slp). Such functions are of vital importance in describing the anisotropy of the electronic charge distribution in the vicinity of the lone pairs and bonds being made and broken.62 As in the earlier study, first-order CI wave functions were used in omjunction with the itcrative natural orbital method6' for determining a nearly optimum set of molecular orbitals. Table I1 gives a brief summary of the 1972 theoretical p r e dictions for F + H2. The a m e n t between the predicted barrier height (1.66 kcal)61and the experimental activation energy (1.71 kcal)u was remarkable. However, BOPS noted soberly that, "the apparently perfect agreement with experiment for the CI harrier height is probably fortuitous. The experimental activation energy may he too low, and the harrier height is not the same as the experimental activation energy, in any case". Furthermore, the error of 2.6 kcal for the exothermicity made it clear that all was still not perfect with the energetics. After the publication by BOPS, another good omen concerning the reliability of that potential surface appeared. Muckerman (61) C. F. Bender, S.V. 0"eil. P. K. Pearson. and H. F. Schaefer, Science, 176, 1412 (1972). (62)T. H. Dunning, 1. Chem. P h p . , 55, 3958 (1971). (63)C.F. Bender and E.R.Davidson, J. Phys. Chcm., 70,2675 (1966).
Figure 3. A view from the exit channel of the 1972 BOPS potential surface for collinear F + H,. Although thwretical refinements of this surface have w u m d since 1972,such refiernents have negligible effsts on this pictorial representation.
I RH.F
2
(%I
Figure 4. Traditional contour map of the I972 BOPS potential energy
surface for collinear F + HI.
had carefully calibrated the parameters of a semiempirical LEPS potential energy surface (PES) by comparing classical trajectory calculations with the observed dynamics. The transition state for this surface (Muckerman V)M occurs at R(F-H) = 1.54 8, and R(H-H) = 0.76 a position identical with that predicted ab intio by BOPS. Since the saddle point position is perhaps the most critical PES feature not directly accessible to experimental determination, this concurrence was significant. Several s u k q u e n t classical dynamics studies using the Muckerman V surface were in Dart iustified hv the aualitative aereement between BOPS and M;ck&man V. ' The entrance and exit channels of the BOPS wtential surface are seen in Figures 2 and 3, and a contour map of the collinear surface is given in Figure 4. As we shall see in following xctions, some specific corrections to the BOPS surface have been made over the past 13 years. However, with the exception of the contour at -34 kcal, Figures 2-4 are essentially indistinguishable, to the naked eye, from what we now consider to be the best representation of the exact PES. In this sense the 1972 work of BOPS brought
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(64) P. A.
(I 974).
Whitlock and 1. T. Muckcrman, J. Chem. Phys., 61, 4624
5340 The Journal of Physical Chemistry, Vol. 89, No. 25, 1985 us a giant step closer to the true potential surface for the F + H2 reaction. BOPS also presented a minimum-energy path or reaction coordinate for F H2. Along this reaction coordinate, the H-H separation increases only slightly until the saddle point or transition state is reached. Further along the reaction coordinate the H-H distance increases and the energy drops steadily, as seen particularly in Figure 2. This presentation of the reaction coordinate in tabular form, as well as an earlier such tabulation6safor H3, anticipated later directed to general definitions of reaction paths for chemical reactions. Are there any simple models for relating a PES such as that of BOPS to the e ~ p e r i m e n t a l ~vibrational *~~ energy distributions for F + H2? Polanyi and co-workers60have developed methods for classifying exothermic A BC potential surfaces as attractive, mixed, or repulsive. An attractive (or early downhill) surface will often yield a degree of vibrational excitation in the AB product molecule, while a repulsive (or late downhill) surface may convert more of the exothermicity into product translational energy. This simple picture becomes more complicated when mixed energy releasea is considered. Qualitatively, a surface is termed attractive if the exothermicity is released as the A atom approaches the BC molecule. In the same way, the surface is repulsive if the energy is released as AB and C separate. Mixed energy release is said to occur if the AB and BC internuclear separations simultaneously change as the energy is released. BOPS concluded that a reasonable characterization of their ab initio PES was provided by Polanyi’s “minimum path method”. With this approach the BOPS surface is characterized as 23% attractive, 22% mixed, and 55% repulsive. This simple model provides a satisfactory correlation with the experimental finding that about two-thirds of the exothermicity for reaction 1 becomes product H F vibrational excitation. Other models considered were unrealistic in that taken literally they would imply no vibrational excitation of the product molecule.
+
+
+
The H FH Exchange Reaction: An Unqualified Success The importance of semiempirical potential energy surfaces for systems such as F H2 is underlined by the impossibility of doing ab initio calculations on all possible geometries. Therefore, semiempirical methods such as LEPS should be viewed as ways to extend (via least squares or other fitting procedures) a modest number of ab initio points to the entire surface. In this light, we undertook in 1975 a test of 13 semiempirical PES for the simple exchange reaction H FH HF + H (9)
+
+
-
which occurs on a different part of the FH2 energy hypersurface. Since essentially nothing was known from experiment concerning reaction 9 and theory had established some credibility for F + H2, this appeared to provide a good test of the different semiempirical methods. A slightly larger basis set than that of BOPS was adopted by Bender, Garrison, and this author (BGS)66for the theoretical study of H FH. The new basis set may be designated F(5s3pld), H(3slp). In addition, a more complete treatment of the correlation problem was undertaken, including all single and double excitations of the valence-shell electrons (all but the fluorine 1s-like S C F orbital). A total of 1583 configurations differing by one or two electrons from the S C F reference configuration (7) were thus included. Table 111 gives a comparison between the 13 semiempirical PES and the ab initio predictions of BGS. S C F theory predicted the barrier to be 68 kcal, C I predicted 49 kcal, and BGS estimated that the true barrier would not be less than 40 kcal. The gulf between this ab initio prediction and the semiempirical methods is immense and readily visible. Only Thompson’s LEPS surface
+
(65) (a) D. G. Truhlar and A. Kuppermann, J . Am. Chem. SOC.,93, 1840 (1971); (b) K. Fukui, S. Kato, and H. Fujimoto, J . Am. Chem. Soc., 97, 1 (1975); (c) H. F. Schaefer, Chem. Br., 11, 227 (1975). (66) C. F. Bender, B. J. Garrison, and H. F. Schaefer, J . Chem. Phys., 62, 1189 (1975).
Schaefer TABLE III: Barrier Height and Saddle Point Geometry for H -HF+H”
R(H-F),
type of potential energy surface
A
authors
BEBO‘
Johnston” Muckermanb I Jaffe and Anderson‘ LEPS Muckermand I1 Muckerman 111 Muckerman IV LEPS Wilkins‘ LEPS Thompson‘ semiempirical valence bond Blais and Truhlarg diatomics in molecules Tullyk I Tully I1 LEPS Muckerman’ LEPS Polanyi and Schreiber’ a prior methods BGSk self-consistent field configuration interaction BGS estimated LEPS” LEPS
+ FH
barrier, kcal/mol
1.10 1.04 1.04
6.8 1 .o -5.2
1.04 1.05 1.05 1.04 1.12 1.10
1 .o 1.7 2.3 1.4 28.6 14.0
1 .os 1.09 1.04 1.05
14.4 13.1 1.2 3.5
1.12 1.14
67.8 49.0 240
“Reference 85. b J . T. Muckerman, J . Chem. Phys., 54, 1155 (1971). ‘R. L. Jaffe and J. B. Anderson, J . Chem. Phys., 54, 2224 (1971). d J . T. Muckerman, J . Chem. Phys., 56, 2997 (1972). ‘R. L. Wilkins, J . Chem. Phys., 57, 912 (1972). fD. L. Thompson, J . Chem. Phys., 57, 4170 (1972). gN. C. Blais and D. G. Truhlar, J . Chem. Phys., 58, 1090 (1973). J. C. Tully, J . Chem. Phys., 58, 1396 (1973). i Reference 64. ’Reference 41. kReference 66. ‘Bond energy, bond order. London-Eyring-Polanyi-Sato. “The saddle point is assumed to occur for a linear symmetric H-F-H geometry.
gives plausible agreement with the more rigorous theory-all 12 other methods differ by more than 25 kcal from the best ab initio estimate. Although the ab initio procedures of BGS66 for this constrained67 barrier height seemed straightforward, there were questions raised6* as to whether the semiempirical methods, predicting much lower barriers, might be correct after all. The high ab initio barrier was inferred by not a few scientists to be inconsistent with the low experimental barrier69 ( 5 8 kcal) for the analogous chlorine reaction
H
+ C1H
-
HC1 + H
(10)
+
However, two kinetic s t ~ d i e s ’ ~of * ~the ’ H FH exchange appeared shortly thereafter and, although not demanding a high barrier, could be interpreted in a manner consistent with the BGS predictions. The resolution of the H + FH barrier height problem was provided by Polanyi‘s group at Toronto in 1978. Using the method of chemiluminescence depletion with mass spectrometry (CDMS), they studied the reaction of vibrationally excited H F with atomic hydrogen. Bartoszek, Manos, and P ~ l a n y i ’determined ~ that the barrier height for reaction 9 must be substantially in excess of 33.5 kcal, the activation energy for the abstraction reaction H
+ FH
+
H2
+F
(1 1)
They conclude, “It is correspondingly encouraging that the ab
initio calculations yielded this result well in advance of the experimental observation”. The final act of the H FH drama would be provided by the resolution of the H + C1H activation energy. Admittedly, it would appear to violate the integrity of the periodic table if the H + FH
+
(67) Although unrelated to the comparison with semiempirical results, the HFH PES turns out to be very flat with respect to bending. See W. R. Wadt and N. W. Winter, J . Chem. Phys., 67, 3068 (1977). (68) D. R. Herschbach, personal communication, 1976. (69) R. E. Weston, J . Phys. Chem., 83, 61 (1979). (70) R. F. Heidner and J. F. Bott, J . Chem. Phys., 63, 1810 (1975). (71) J. F. Bott, J Chem. Phys., 65, 1976 (1976). (72) F. E. Bartoszek, D. M. Manos, and J. C. Polanyi, J . Chem. Phys., 69, 933 (1978).
The Journal of Physical Chemistry, Vol. 89, No. 25, 1985 5341
Feature Article
+
and H ClH barriers were 40 and 5 kcal, respectively. However, high-quality theoretical prediction^^^-^^ have for the past 7 years insisted that the true H CIH exchange barrier is closer to 20 kcal. The two most recent laboratory s t ~ d i e sof~ H ~ ? ~ClH ~ contradict the earlier experiment^^^ and do indeed appear consistent with a barrier near 20 kcal for reaction 10.
+
+
The Agony Begins: 1977 In 1973 it became apparent to Bender and this author that extension of the BOPS61approach to a larger basis set would not improve the caliber of the PES. Specifically, it was found that the use of the first-order C I approach with a larger basis set increased the exothermicity to -38 kcal, in poor agreement with the experimental 3 1.8 kcal. To make matters worse, a more complete description of the correlation problem, the inclusion of all single and double excitations (CISD, as reported by BGS for H FH), increased the predicted barrier to an unacceptably high value. Thus, it was established by 1975 that the BOPS treatment represented a “Pauling point”, a serendipitous level of theory beyond which matters deteriorated initially rather than improving. Such a possibility, as we have noted, had been anticipated by BOPS. Still desirous of finding the perfect theoretical potential energy surface for F + H2, Ungemach, this authors, and Liu (USL)78 engaged in a very ambitious (for 1976), high-level theoretical assault on the problem. Their results were presented in September of 1976 during the Faraday Society Discussion on Potential Energy Surfaces, at the University of Sussex. USL used a basis set more than twice the largest previously applied to the F H2 problem. Their basis may be designated F(6s4p3dlf), H(3s2pld). In fact, even now, 9 years later and after great improvements in theoretical methods4’ and computer t e c h n ~ l o g ythe , ~ ~USL basis is still a large one for the FH2 system. It should go without saying that essentially all of the classic “difficult” pr0b1em.s~~ in quantum chemistry have been solved with smaller basis sets than that used by USL. USL were also the first to apply the multireference CI methods0 to the F + H2 problem. For example, a frequent starting point in their research was the three-configuration SCF wave function
+
+
+
+
\k = C, la22a23a24ala4 C2 la22a24a5a21a4 c 3
la~2o23o4~5ol?r4 (12)
Then single and double excitations (a total of 6874 configurations) with respect to all three reference functions were included in the CI. With this three-reference CI, the collinear transition state was explicitly located at R(F-H) = 1.48 %, and R(H-H) = 0.78 A. Given the quality of the wave functions used, all subsequent calculations assumed this theoretical transition state. With their large basis set, USL predicted the one-reference CISD barrier for F H2 to be 6.0 kcal, very much higher than the experimental activation energy of 1.7 kcal. The more complete three-reference CI reduced the ab intio barrier to 3.9 kcal, still much greater than experiment. It is apparent that the triple and quadruple excitation^^^,^^ included in the latter theoretical treatment decrease the barrier height by 2.1 kcal, a significant amount. This was perhaps the first explicit demonstration of the importance of such higher excitations on predicted barrier heights. The largest MCSCF wave function considered by USL was the eight-configuration wave functiong1
+
(73) P. Botschwina and W. Meyer, Chem. Phys., 20, 43 (1977). (74) T. H. Dunning, J . Chem. Phys., 66, 2752 (1977). (75) A. F. Voter and W. A. Goddard, J . Chem. Phys., 75, 3638 (1981). (76) J. C. Miller and R. J. Gordon, J . Chem. Phys., 76, 5167 (1982). (77) C. A. Wight, F. Magnotta, and S. R. Leone, J . Chem. Phys., 81,3951 (1984). (78) S. R. Ungemach, H. F. Schaefer, and B. Liu, Faraday Discuss. Chem. SOC.,62, 330 (1977). (79) See,for example, V. R. Saunders and J. H. Van Lenthe, Mol. Phys., 48, 923 (1983). (80) P. S. Bagus, B. Liu, A. D. McLean, and M. Yoshimine, ”Wave Mechanics: The First Fifty Years”, W. C. Price, S. S . Chissick, and T. Ravensdale, Eds., Butterworths, London, 1973, pp 99-1 18.
* = C, 3a24al?r4 + C2 4a5a21?r4+ C3 3a4a5aln4 (spin cs 3
+
+
coupling 1) C4 3a4aSa1x4 (spin coupling 2) ~ ~ c 56 3~ ~1 4~ c ~7 ~ 4~ ~1 ~~ cs 5~ ~
+
+
+
1 ~ (13) This MCSCF wave function is a full three-electron, three-orbital CI; Le., it includes all possible arrangements of three electrons in the three atomic orbitals 2pu(F), ls(H,), and ls(HB). This means, among other things, that MCSCF wave function (1 3) is a rigorous theoretical implementation of the valence-bond ideas behind the LEPS model.60 (In fact, the eight-configuration MCSCF treatment is more complete than the valence-bond wave function that underlies the LEPS model, since the latter includes only two linearly independent configurations that correspond to covalent bonding schemes for arranging three electrons in three orbitals.) Including all singly and doubly excited configurations with respect to all eight reference configurations provided a further lowering of 0.58 kcal to the three-reference C I barrier. At this point, the agony of the situation was intensely felt by USL. Their best ab initio based prediction of the F + H2 barrier height was 3.35 kcal, still much higher than the 1.7-kcal experimental activation energy. Their response was to take a deep breath and state,78“It seems improbable that our predicted barrier is more than 1 kcal larger than the exact value”. Thus, USL suggest 2.35 kcal as a lower limit for the classical barrier height for the F H2 reaction. Although we are not jumping ahead a bit, one may state that, 9 years later, no higher level of ab initio theory has yet to challenge this statement by USL.
+
Continuing Agony and a Glimmer of Hope: 1984 In the years between 1976 and 1983 the F + H2PES continued to be examined at Berkeley. CI procedures were examined (for example, including all triple and quadruple excitations with small basis sets), occasionally tested, and discarded. In 1983 it was decided that an alternative theoretical approach might provide a new window into the FHz problem. Specifically, the use of fourth-order perturbation theorys2 was chosen by Frisch, Binkley, and this author (FBS).83 Fourth-order perturbation theory, or MP4 (an abbrevation standing for fourth-order Maller-Plesset perturbation theory), takes explicit account of the effects of all triple and quadruple excitations, Le., all configurations differing by three or four electrons from the S C F reference configuration (7). However, the fourth order of perturbation theory is the earliest order in which single, triple, and quadruple excitations appear. Thus, one may say that MP4 treats single, triple, and quadruple excitations in a manner roughly comparable to the way double excitations are treated in second-order perturbation theory. This analysis highlights the differences with multireference CI, which treats the most important triple and quadruple excitations in a manner equivalent to some very high order of perturbation theory. We consider the two methods (MP4 and MRCISD) to be complementary and in the past have applied both methods to particularly difficult chemical problems.84 FBS employed a wide range of basis sets, from less than double-!: (DZ) caliber to near-Hartree-Fock in the valence-shell description. Their largest basis set, designated 6-3 1 1++G(3df,3pd), may also be labeled F(5s4p3dlf), H(4s3pld). Such a basis set is roughly comparable to that used by USL several years earlier78and is far more than adequate for the definitive theoretical solution of most chemical problems. A secondary result of the work of FBS is the establishment of an estimate for the Hartree-Fock limit for the classical barrier height. The FBS classical or “bare” barrier height with their largest basis set is 16.1 kcal, which is certainly within 1 kcal of the exact (unknown) (81) The MCSCF method used by USL was that of J. Hinze, J . Chem. Phys., 59, 6424 (1973). ( 8 2 ) R. Krishnan, M. J. Frisch, and J. A. Pople, J . Chem. Phys., 72, 4244 (1980). (83) M. J. Frisch, J. S. Binkley, and H. F. Schaefer, J . Chem. Phys., 81, 1882 (1984). (84) K. S. Kim, H. F. Schaefer, L. Radom, J. A. Pople, and J. S. Binkley, J . Am. Chem. SOC.,105, 4148 (1983).
~
5342
The Journal of Physical Chemistry, Vol. 89, No. 25, 1985
Hartree-Fock result. This provides a quantitative measure of the failure of Hartree-Fock theory for F H2. FBS also discussed from the point of ab initio theory the relationship between the classical barrier for the F + H 2 reaction and the expected activation energy. The classical barrier, of course, is the energetic difference between the F H2minimum structure and the F-H-H transition-state geometry. The classical barrier Ea takes no account of the vibrational degrees of freedom accompanying the reaction. The simplest treatment of zero-point vibrational energy (ZPVE) is provided by ordinary, garden-variety transition-state theory.85 In this simple picture one predicts for the activation energy
+
+
E, = Ec, - ZPVE(reactant)
+ ZPVE(transition state)
(14)
The highest level of theory at which FBS predicted vibrational frequencies was second-order perturbation theory (MP2) using a 6-31 1G** basis set, essentially equivalent to double-{ plus polarization (DZ p). Furthermore, the vibrational analysis was restricted to the harmonic approximation.86 Moreover, in simple transition-state theory the imaginary vibrational frequency (corresponding to the reaction coordinate) is ignored in the determination of changes in ZPVE. In such a picture, FBS predicted F + H2 to have 1.44 kcal/mol more ZPVE than does the transition state. This means that
+
E , = Ec, - 1.44 kcal/mol
(15)
Such a relationship, of course, were it valid, significantly changes the presumed discrepancy between theory and experiment for the activation energy of F H2. The lower limit theoretical estimate of 2.35 kcal by USL would be translated into an activation energy of 0.91 kcal/mol. Such a prediction is of course in agreement with the two latest experimental activation energies, namely 0.86 f 0.10 and 1.18 f 0.10 kcal. Thus, there appeared hope for an eventual concurrence between theory and experiment. The highest level of theory considered by FBS was fourth-order perturbation theory using their F(5s4p3dl0, H(4s3pld) basis set. The F H2 classical barrier predicted at this level of theory was 3.24 kcal, only slightly less than USL’s earlier of 3.35 kcal. The essential agreement between two rather different approaches to the correlation problem gave credence to the theory’s prediction of a relatively large classical barrier for the F + H, reaction.
+
+
Theory’s Latest Stand The most recent efforts on the F H 2 potential surface combined the forces of the research groups at IBM San Jose, Sandia Livermore, and Berkeley. Both multireference (MR) CISD (configuration interaction including all single and double excitations)80 and MP4 (full fourth-order perturbation theory)82 methods were used in this latest theoretical assault. One criticism of the previous theoretical studies of USL7*and FBSs3 would be that the stationary-point geometries (reactants, transition state, products) were not located at the highest level of theory considered. The work of Frisch, Liu, Binkley, this author, and Miller (FLBSM)87 removes this deficiency of the earlier theoretical studies and presses on to even higher levels of theory. The work of FLBSM must certainly be considered state of the art for ab initio quantum chemistry in 1985. Following the order of presentation of FLBSM, we describe the MRCISD theoretical approach first. A very large basis set, designated F(7s5p3d20, H(4s3p2d), was used, differing from the 1976 USL by the addition of s, p, and f functions on fluorine and s, p, and d functions on hydrogen. Molecular orbitals were determined via an MCSCF procedure which included all configurations qualitatively arising from F 2p, Ha Is, and H b I s atomic orbitals. This complete active space self-consistent-field
+
(85) H. S. Johnston, ‘Gas Phase Reaction Rate Theory”, Ronald Press, New York, 1966. (86) E. B. Wilson, J. C. Decius, and P. C. Cross, ”Molecular Vibrations”, McGraw-Hill, New York, 1955. (87) M. 1. Frisch, B. Liu, J. S. Binkley, H. F. Schaefer, and W. H. Miller, Chem. P h y ~ L. e t t . . 114, 1 (1985).
Schaefer TABLE I V Predicted Saddle Point Geometries for the F FH + H Reaction
year 1972 1972 1974 1976 1984 1985
-
R(F-H),
WH-H),
1.54 1.54 1.43
0.767 0.76 0.777
1.48 1.36 1.44 1 .47a 1.48
0.778 0.780 0.77 0.77 0.77
A
authors/method BOPS61/first-orderCI Muckerman V64/semiempirical P~lanyi-Schreiber~~ SE-1/ semiempirical USL78/three-referenceCI FBSs3/MP2 6-31 1G** FLBSMs7/MPA FLBSM/multireference CI FLBSM/above with g functions
+ H2 A
“Note that ref 87 contains a typographical error for this number. TABLE V: Predicted Classical Barrier Heights for the F FH H Reaction
+
+ H2
-
ECk
year
authors
method three-reference CI above corrected for eight-reference effects fourth-order perturbation theory 1984 FBSS3 1985 FLBSMg7 fourth-order perturbation theory multireference CI above with g functions 1976 USL7*
kcal/ mol 3.93 3.35 3.24 3.68 3.43 3.24
(CASSCF)88 wave function is an improvement on the eightconfiguration MCSCF wave function of USL, since the latter constrained the fluorine 2pr-like orbital to hold four electrons in all configurations. Specifically, the additional configurations in the CASSCF wave function are 1 2 2 u23 u24a25u 1r
2
1U22u23u4u25 u21 r
2
(16)
The MRCISD procedure then amounted to including all single and double excitations with respect to all eleven reference configurations (1 3) and (1 6), plus reference functions involving zero or one electron in the 2u (fluorine 2s-like) molecular orbital. Furthermore, a second basis set was used, in which a set of nine (1 = 4; ml = 4, 3, 2, 1,0, -1, -2, -3, and -4) g functions was added to the above, in a completely analogous MRCISD procedure. With both basis sets, the F + H2 transition state was located by using the full MRCISD wave functions. The four-order perturbation treatment explicitly located the stationary points using the largest FBS basis set, namely F(5s4p3dlf), H(4s3pld). In addition, given these three stationary points, single-point calculations were performed using a large, uncontracted Gaussian basis set, designated F( 1 ls7p5d30, H(7s5p3d). With the exception of the omission of g functions on fluorine, such a basis set is approaching valence-shell completeness rather closely. The predicted saddle point geometries are compared with earlier work in Table IV. As noted earlier, the BOPS6’ and Muckerman V64 surfaces are in close agreement concerning the transition-state geometry. A second (after Muckerman V) well-known semiempirical potential energy surface for F + H2 is the SE- 1 surface of Polanyi and S~hreiber.~’ Like Muckerman V, the SE-1 surface was designed to yield classical trajectory simulations of the dynamics in good agreement with experimental finding^.^^-^^ Table IV indicates concurrence of the highest levels of theory that the H-H distance at the transition state is 0.77 A, confirming the early ab initio prediction of BOPS. For the F-H distance, the agreement among theoretical methods is less precise, with fourth-order perturbation theory giving 1.44 A and multireference CI giving 1.48 h;. Nevertheless, it can be stated that there is (88) B. 0. Roos, P. R. Taylor, and P. E. M. Siegbahn, Chem. Phys., 48, 157 (1980); K. Ruedenberg, M. W. Schmidt, M. M. Gilbert, and S. T. Elbert, Chem. Phys., 71, 41 (1982).
The Journal of Physical Chemistry, Vol. 89, No. 25, 1985 5343
Feature Article
TABLE VI: Predicted Vibrational Frequencies (in cm-I) and ZPVE Corrections to the Classical Barrier for F + H2 v(reaction method coordinate) u( bend) v(H-H) MP2/6-31 IG** MP2/F(5s4p2d), H(4s2p) MP4/F(Ss4p3dlf), H(4s3pld) MRCI/F(7s5p3d2f), H(4s3p2d) MRCI/F(7sSp3d2flg), H(4s3p2d)
1380i 12571' 931i 81 l i 819i
generally good agreement between the highest levels of theory for the F + H2 saddle point geometry. The 6-31 1G** second-order perturbation theory (MP2) F-H distance in Table IV is 0.08-0.12 b; shorter than the higher level theoretical results. However, FLBSM showed that the use of a larger basis set, namely F( 5s4p2d), H(4s2p), increases the MP2 F-H distance to 1.40 A. Therefore, the inherent error in second-order perturbation theory is not as large as might appear to be the case from the small-basis result in Table IV. Recent theoretical values of the classical barrier height for F H2 are given in Table V. A brief explanation of the two fourth-order perturbation theory results in Table V is in order. The two results (1984 and 1985) actually came from the same basis set, the F(5s4p3dlf), H(4s3pld) basis of FBS. The difference is that the former barrier (3.24 kcal) assumed stationary-point geometries predicted at a lower level of theory, while the latter barrier (3.68 kcal) is a completely consistent theoretical result. Table V suggests that the theoretical barrier height may be converged at -3.2 kcal/mol. Since g functions lower the barrier by only 0.2 kcal, it is not apparent that further extensions of the basis set will accomplish much additional lowering. However, in preliminary studies, Liu has vastly increased the size of the CASSCF reference function by adding the fluorine 3p virtual orbital to the active orbitals. With a multireference CI based on this very large MCSCF wave function, some lowering (0.8 kcal is suggested by FLBSMa7)of the barrier occurs, when stationary-point geometries are assumed from lower levels of theory. However, the MP4 results with a nearly saturated F(spdf), H(spd) basis set suggest some increase in the predicted barrier height. We must conclude by falling back on the 1976 estimate of USL that 2.35 kcal is a lower limit to the classical barrier height.
+
Concluding Remarks: Zero-Point Vibrational Energy (ZPVE) and Tunneling In light of the failure to obtain a classical barrier height for F H2 in agreement with the Arrhenius activation energy ( 1 k ~ a l ~ ' ,questions ~ ~ ) , concerning the ZPVE become more important. Table VI summarizes the theoretical predictions to date for the saddle point harmonic vibrational frequencies. There it is seen that these frequencies are quite sensitive to the level of theory employed. At the highest level of theory for which the bending frequency was evaluated, namely MP4, v(bend) becomes very small, 66 cm-'. Such a result is, however, consistent with the earliest ab initio study, that of BPOSS4 The H-H stretching frequency at the saddle point is seen in
+
N
184 277 66
3158 3284 3463 3497 3455
AZPVE,
kca I / mol -1.44 -0.95 -1.19
Table VI to be essentially converged at -3460 cm-'. The MP4 (3463 cm-I) and largest basis MRCI (3455 cm-') results differ by only 8 cm-'. For the purpose of evaluating the ZPVE of reactants and transition state within the harmonic approximation, the large basis MP4 results would appear adequate. Again, the reader should be reminded that this is the crudest reasonable treatment of the effects of ZPVE on the course of the reaction. Fourth-order perturbation theory (MP4), in conjunction with simple transition-state theory, suggests that the Arrhenius activation energy is 1.19 kcal less than the classical barrier height. Substracting this 1.19 kcal from the 3.24-kcal classical barrier (last line of Table V) gives 2.05 kcal for E,. FLBSM also found that the simplest treatment of tunneling within the framework of transition-state theorya9 provides a further lowering of the threshold for chemical reaction. FLBSM suggested that a classical barrier of 2.7 kcal (0.54 kcal below their strictly ab initio result) would be consistent with the experimental activation energies2'sZa and molecular beam reaction threshold.],* Just as rapprochement with experiment appeared possible, a more detailed theoretical treatment of the dynamics appeared. Steckler, Truhlar, and Garrett" assumed a semiempirical potential energy surface in conjunction with the use of variational transition-state theory. They concluded that the use of ordinary transition-state theory gives too large a correction (to the classical barrier) for the effects of ZPVE. They further state that the effects of anharmonicity and multidimensional tunneling conspire to yield an activation energy rather close to the classical barrier height. To summarize, theoretical efforts of the past decade to characterize the F H2 potential surface continue to suggest an ab initio lower limit of 2 kcal for the classical barrier height. Problems appear to remain in reconciling this theoretical prediction with experiment. Thankfully, we know of no other chemical problem for which such high levels of theory conflict with e ~ p e r i m e n t . ~ ~
+
Acknowledgment. I am deeply appreciative of the superb research efforts provided by my colleagues Charles F. Bender, Peter K. Pearson, Stephen V. O'Neil, Steven R. Ungemach, Bowen Liu, Michael J. Frisch, J. Stephen Binkley, and William H. Miller. This research was supported over the past 15 years by the U S . Department of Energy and the U S . National Science Foundation. Registry No. F, 14762-94-8; H2, 1333-74-0. (89) W. H. Miller, J . Am. Chem. SOC.,101, 6810 (1979). (90) R. Steckler, D. G. Truhlar, and B. Garrett, J . Chem. Phys., 83,2870 (1985).
