5298
J. Phys. Chem. 1994,98, 5298-5302
Potential Energy Surfaces and Dynamics of Proton-Transfer Reaction 0-
+ HF
-
OH( v)
+ F-
Hiroto Tachikawa,' Hiroshi Takamura, and Hiroshi Yoshida Faculty of Engineering, Hokkaido University, Sapporo 060, Japan Received: January 20, 1994; In Final Form: March 15, 1994'
+
-
The gas-phase proton-transfer reaction 0- HF OH(u = 0,l) + F- has been studied with the ab initio MO method and quasiclassical trajectory calculations. A strongly bound intermediate complex [OHFI- is found on the ground-state potential energy surface (PES) obtained by the ab initio MO method. The intermediate complex is most stable a t the collinear form. Three-dimensional quasiclassical trajectory calculations a r e performed with ab initio fitted PESs. The results show that the enhanced collision energy from 1.198 to 4.10 kcal/mol increases the product OH(v = 1) fractional population, P(u = 1) = (OH(u = l)/[OH(o = 0) OH(u = l)]). The theoretical result suggests that this increase is due to the energy transfer from the translational mode to the vibrational mode of 0-H in the deeper intermediate complex region. The trajectory calculations show that P(u = 1) a t a collision energy of 5.31 kcal/mol is slightly smaller than that of 4.10 kcal/mol. These results a r e in reasonable agreement with experimental features derived by Leone and co-workers [ J . Chem. Phys. 1992, 96, 2981. O n the basis of the theoretical calculations, we propose a reaction model composed of two reaction channels: one is a n intermediate complex channel model in which the reaction proceeds via a long-lived intermediate complex, [OHFI-, and the other is a direct channel model in which the reaction proceeds directly without the long-lived complex. The direct channel gives the vibrationally excited OH(u = 1, J = 0) radical, whereas the complex channel leads to the vibrationally ground and rotationally excited OH(u = 0, J = J 3 radical.
+
1. Introduction
Proton transfer, hydrogen atom transfer, and charge transfer are the most fundamental chemical elementary processes which are observed widely in the gas phase as well as in the condensed phase.' Proton transfer in the gas phase is one of the simplest reactions, including bond-forming and bond-breaking processes. Therefore, the reactions have been extensively studied for elucidation of the nature of chemical reactions.* The protontransfer reaction in the heavy-light-heavy system
A-
+ H-B
+
A-H
+ B-
(1)
where A and B are heavy atoms, has been investigated by means of the crossed molecular beam technique,' drift and selected ion flow tubes t e c h n i q ~ e ,infrared ~ e m i ~ s i o n ,and ~ laser-induced fluorescence (LIF)techniques6 These experimental studies show that (i) the relative translational energy between the products (A-H and B-) increases with increasing collision energy, and (ii) the energy transfer from translational mode to vibrational mode of the product does not occur effectively. Recently, an exceptional reaction was found by Knutsen et al.' They determined, using a flow-drift tube technique, the vibrationalstate populations of the product O H radical formed by a protontransfer reaction, 0-+ H F + O H ( u = 0 , l )
+F
(1)
The fractional ratio of product OH(u = 1)
P(v = 1) =
OH(u = 1) OH(u = 0) + OH(u = 1)
(2)
increases monotonically with increasing collision energy. Thus the kinetic feature of this reaction is much different from the previously observed features. In the present study, the potential energy surfaces of reaction I are calculated by means of the ab initio M O method, and the OAbstract published in Advance ACS Abstracts, April 15, 1994.
0022-365419412098-5298%04.50/0
vibrational and rotational distributions of the product O H are determined by using the quasiclassical trajectory calculations on the ab initio fitted PESs. Primary aims of the present study are (i) to provide theoretical information on reaction I and (ii) to discuss the mechanism of the proton-transfer process in reaction I on the basis of both the PES characteristics and the results obtained from the quasiclassical trajectory calculations. In the following section, the method of the calculations is described. In section 3, the ab initio MO calculations are presented. The results of the classical trajectory calculations on the ab initio fitted PESs are also shown. The conclusion and discussion are described in section 4. A new theoretical explanation of reaction I will be also described in summary.
2. Method of Calculations A. Ab Initio MO Calculations. Geometries for the neutral HF molecule, the OH radical, and the reaction intermediate [OHFI- are fully optimized by means of the energy gradient methods with 4-31G*, 6-31G*, 6-31G**, and 6-31++G** basis sets.9 The electron correlation energy for each geometry is estimated by the double-substituted configuration interaction method (D-CI)Ioa and Merller-Plesset second- and third-order perturbation methods (MP2 and MP3)" within the frozen-core approximation with the 6-3 1G** and 6-3 1++G** basis sets. Unlinked cluster quadruple correction (QC)is added to allow for the size-consistency correction.10b-d Adiabatic potential energy surfaces (PESs) for reaction I as functions of the 0-H and H-F distances are obtained a t the Hartree-Fock (HF) level with 6-31++G** basis sets. Two sets of 0-H-F angles (e), 180 and 135O, are chosen to express the angle dependency. The ab initio M O calculations are carried out at 120 points on the PES. B. Quasiclassical Trajectory Calculations. The PESs obtained theoretically are fitted to an analytical equation by means of the least-squares method. The extended LEPS surface is employed to express the analytical function of the PES; 2000 trajectory calculations on the LEPS surface are computed at each initial condition (HF(u = 0, J = 0 ) and collision energies (EW1l~Jof 1.198, 3.00, 3.68, 4.10, and 5.34 kcal/mol). Integration of the 0 1994 American Chemical Society
Proton-Transfer Reaction 0- + H F
-.
OH(u)
+ F-
The Journal of Physical Chemistry, Vol. 98, No. 20, I994 5299
TABLE 1: Optimized Geometries of HF, OH, and Intermediate Complex [OHF]-. Bond Lengths and Angles in Angstroms and Degrees, Respectively
O'+HF
-\
[OHFImethod
r(H-F)
HF/4-31G HF/6-31G* HF/6-3 1G** HF/6-31++GS* MP2/6-3 1++G**"
0.9006 0.9022
r(O-H)
0.9549 0.9549
r(H-F) 1.2981 1.2824 1.2608 1.4003 1.35
r(O-H) 1.1001 1.0848 1.0946 1.0314 1.08
0 180.0 180.0 180.0 180.0 180.0
From ref 14.
classical equations of motion is performed at the standard fourthorder Runge-Kutta and sixth-order Adams Moulton combined algorithm1*with the time increment of 1 X s. C. Lifetime of Intermediate Complex [OHFt (ZII). The lifetime (7) of the [OHFI- intermediate on the ground state 2DPES is estimated by using RRK theory.l3 The lifetime as a function of energy E is expressed by
-40.
+
-
+
Figure 1. Energy diagram of the 0- H F OH F- reaction. The numbers in parentheses are the HF/D-CI+QC energies relative to the reactant scale calculated with the 6-31++G** basis set.
4.0
.
3.0
a
,hl
2.0
where ( v ) is an average of vibrational frequencies of modes related to thereaction, Vis thedissociation energy of thereaction [OHFIOH F-, and s is the number of degrees of freedom of vibrations. ( v ) is calculated with the OH stretching mode and the H F stretching mode of [OHFI- obtained at the MP2/631++G** level.
1.o
+
4.0
3. Results
A. Global Featuresof the Proton-TransferReaction. To obtain the structures of reactant, product, and an intermediate [OHFI-, geometry optimizations are performed. The geometries obtained are listed in Table 1. Four independent calculationsgive a similar geometry. The 0 - H and H-F bond lengths of the intermediate complex [OHFI- are 1.0314 and 1.4003 A in the 2I-l state, respectively. The H-F distance is much larger than the 0 - H distance, and the skeleton of 0-H-F- is most stable for a linear form (0 = 180'). These features are similar to the previous theoretical re~u1ts.l~ Total energies, heats of reaction (AH), and the complex formation energies ( A E ) calculated at several levels of theory are given in Table 2. The heat of reaction calculated with HF/631++G** basissets (13.19 kcal/mol) isessentiallyinaccordance with the experimentalvalues (10.6 kcal/mol).I5 This agreement implies that one can discuss the reaction mechanism with the HF/6-31++G** level. Based on the relative energy calculated at the D-CI+QC/63 1++G** level, the energy correlation diagram is illustrated in Figure 1. The present proton-transfer reaction has a strongly bound intermediate complex [OHFI- at the collision region. This feature is in good agreement with the prediction from the MP2/ 6-31++G** 1 e ~ e l . I ~
(-46 5 I - 4 4 6)
-60.
(3)
-
OH t F-
.
3.0
a
cu
L
2.0 1.o 1.o
2.0
3.0
4.0
-
r,I A
+
Figure 2. Potential energy surfaces of the 0- + H F O H F- reaction calculated at the HF/6-31++Gt* level: (A) collinear collision (0 = 180°), (B) coplanar collision (0 = 135O). Contours are drawn for each 5 kcal/mol.
B. Potential Energy Surface. The adiabatic PESs of the zArr state for the 0- H F reaction are shown in Figure 2. A strongly bound complex is formed at r(HF) = 1.30 A and r ( 0 H ) = 1.05 A.on the ground state PES (e = 180O). The potential basin of the intermediate complex [OHFI- is deepest for the collinear collision. The stabilization energy of [OHFI- is calculated to be 1.58 eV with respect to the initial state (r(0---H) = 4.2 A and r(H-F) = 1.0 A).
+
(2I-l)
-
TABLE 2 Total Energies (in au), Complex Formation Energies (AEin kcal/mol), and Heats of Reaction (AH in kcal/mol) for the 0- + HF OH + F- Reaction System method 0HF IOHF1OH FA E A H HF/6-31G**//HF/6-3 1G** DCI/6-31GIS//HF/6-31G**
DCI+QCb/6-31G**//HF/6-31G** MP2/6-3 1G1*//HF/6-31G**
MP3/6-31G**//HF/6-31++G**
HF/6-3 1++GI *//HF/6-3 1++G* * MP2/6-3 1++G**//HF/6-3 1++G* * MP3/6-3l++G**//HF/6-3 l++G** DCI/6-3 l++G**//HF/6-3 l++G** DCI+QCb/6-3 1++G**//HF/6-3 1++G* *
-74.718 701 -74.865 239 -74.869 403 -74.857 755 -74.867 936 -74.766 619 -74.925 248 -74.930 774 -74.926 957 -74.932979
"//" means "optimized by". * Size-consistency correction.
-100.01 1 691 -100.191 823 -100.197 169 -100.194 125 -100.196 021 -100.024 312 -100.215 231 -100.214 503 -100.210 064 -100.215 970
-174.812 -175.134 -175.156 -175.143 -175.153 -174.861 -175.217 -175.219 -175.198 -175.223
Basis eot superposition error
858 877 471 891 941 814 319 374 087 085
-75.388 -75.541 -75,546 -75.531 -75.543 -75.393 -75.540 -75.552 -75.549 -75.554
331 368 020 803 863 364 909 371 628 624
-99.350 -99.523 -99.528 -99.526 -99.527 -99.418 -99.623 -99.613 -99.609 -99.616
482 531 279 607 506 586 847 63 462 117
51.78 48.83 56.41 57.74 56.47 44.6Oe 48.22 46.50 38.32 46.52
5.28 4.92 4.85 4.10 4.65 13.19 15.23 13.00 13.84 13.68
(BSSE)of tho complex formation energy is 0.46 kcal/mol.
Tachikawa et al.
5300 The Journal of Physical Chemistry, Vol. 98, No. 20, 19'94
TABLE 4
Summary of Trajectory Calculations
E,II kcal/mol 1.198 3.00 3.68 4.10 5.34
4ff
P(u = 1)b
Nr" 0.800 0.70 0.60 0.50 0.40
0.14 0.22 0.39 0.43 0.35
0.12 0.18 0.28 0.30 0.26
a Ratio of reactive trajectories, Nr = (number of reactive trajectories)/ (number of total trajectories). OH(u = 1) fractional ratio, P(u = 1) = {OH(v = l)/[OH(u = 0) + OH(v = l)]). Ratio of part A to part B.
05 0.4
l
t
d
\w,
i
I
O
TABLE 3 LEPS Parameters (S, Sat0 Parameter; 8, Morse Parameter in A-1;0, Dissociation Energy in kcal/mol; re Internuclear Distance in A; a, Value in Extended LEPS Parameter) of the ab Initio Fitted PES parameter 0. * .H H. *F 0. *F S 1.47263 1.3243 0.7700 B re (Y
149.12 1.8836 0.9766 1.ooo
135.26 1.9011 0.9234
48.91 1.8144 2.7739
The shape of the PES exhibited by the intermediate complex [OHFI- is strongly and tightly bound along both 0-H and H-F directions. Therefore, it could be expected that energy transfer from the translational mode to the 0-H stretching mode takes place efficiently in this basin by the collision. The PES clearly indicates the vibrational excitation of product OH. C. QuasiclassicalTrajectoryCalculation. Table 3 shows fitting parameters for the PES determined by the least-squares method. The mean energy difference from the ab initio values is less than 1.2 kcal/mol. The well depths, the shape of the PES, and AH are reasonably reproduced by the ab initio fitted PES. Potential energy and interatomic distances versus time for a sample trajectory calculated on the fitted PES are shown in Figure 3. The trajectory with a collision energy of 3.00 kcal/mol falls down the potential basin, and after two collisions with the well, the trajectory leads to product (OH + F-). The elapsed time in the intermediate complex region for this trajectory is about 0.05 ps. The lifetime derived from all trajectory calculations is in the range of 0.05-0.9 ps. Thecorresponding value estimated by RRK theoryIs is 0.22 ps. This means that the excess energy of [OHFIis not completely distributed in the rotational and vibrational modes. Recently, Levandier et presented the angular and kinetic energy distribution for the products of the proton-transfer reaction 0-+ H F OH + F- a t a center of mass collision energy of 9.54 kcal/mol. Their results indicate that the reaction proceeds through a transient [OHFI- complex living several rotational periods. The present theoretical result is consistent with their experiments.
-
4
I
i
Figure 3. Sample trajectory plotted for the potential energy (A) and r(H-F) and r ( G H ) (B) versus time. The trajectory startson theentrance region at time zero and passes rapidly the intermediate complex region.
De
,
'
L
C
0.4
I
$ 0.3 d j $ 0.2 1
0.1
0
2
4
6
a
10
Rotational Quantum Number J Figure 4. Rotational-state populations of the OH radical formed by the 0- + HF OH + F- reaction. The collision energies are (A) 1.198 kcal/mol, (B) 3.00 kcal/mol, and (C) 5.31 kcaI/mol.
-
Rotational Quantum Number J
Figure 5. Schematic representation of the rotational-state population.
D. Analysis of the Trajectory Calculations. Fractional Ratio of the Product OH(v = I ) . The summary of all trajectory calculations is listed in Table 4. The fractional ratio of the product OH(v = 1) defined by eq 2, P(u = l), increases with increasing collision energy from 1.198 to 4.10 kcal/mol. At Ecoll = 5.34 kcal/mol, P(u = 1) is suddenly decreased. This reason will be argued in the discussion part. Rotational-State Populations of the Product OH. Figure 4 shows the rotational-state populations of the product OH calculated at E,II = 1.198, 3.00, and 5.31 kcal/mol. The populations are widely distributed from J = 0 to J = 10. It is clear that the distribution is composed of two components of rotational distribution. A schematic illustration of the two population curves is represented in Figure 5. One component denoted by A has a peak at J = 0, whereas the other component denoted by B peaks a t a higher rotational level ( J = 4-6). The trajectory calculations indicate that part A and part B are
Proton-Transfer Reaction 0-
+ HF
-
OH(u)
+ F-
scattering (rebound scattering). Therefore the long-lived complex channel is dominant a t higher collision energies. B. The Reaction Model. In this section, we propose a reaction model based on the theoretical results. The a b initio potential energy surface (PES) shows that the strongly and tightly bound intermediate complex [OHFI- exists in the collision region on the PES. This deeper well causes the energy transfer from the translational mode to the OH vibrational mode. As described in the previous section, reaction I is composed of two reaction channels. One is a direct channel,
1c
0.4
0-+ HF
t 2 4 6 Collision Energy / kcal/mol Figure 6. Theoretical fraction of the total OH population in u = 1, P(u = l), vs center-of-mass collision energy of the reagents ( 0 ) . The experimental data (0)are cited from ref 7.
0
composed of the vibrationally excited OH(u = 1) and the vibrational ground state OH(u = 0), respectively. From an analysis of each trajectory, it is found that the OH(u = 1) is mainly formed by a direct mechanism (about 90%) and the OH( u = 0) is formed via a long-lived intermediate complex [OHFI-. As shown in Table 4, the ratio A / B increases monotonically with increasing collision energy and peaks at Emll = 4.10 kcal/mol ( A / B = 0.43). At Ecoll = 5.31 kcal/mol, part A is slightly decreased ( A / B = 0.35). 4. Discussion and Conclusion
A. Comparison with Experimental Results. Recently Leone and co-workers7 determined the relative vibrational-state populations for the OH product in the reaction 0- H F OH(u = 0,l) F- as a function of reactant center-of-mass collision energy in a flow-drift tube. At thermal energy, P(u = 1) is obtained as 0.18 f 0.01. At enhanced average collision energies of 2.29 and 3.68 kcal/mol, P(u = 1) increases to 0.25 f 0.02 and 0.33 f 0.03, respectively. For comparison, the present theoretical values are plotted in Figure 6 together with Leone’s experimental results7 Although the present theoretical values are slightly smaller than the experimental values, a collision energy dependency of P(u = 1) is excellently represented by theoretical calculations. The theoretical OH(u = 1) fractional population, P(u = l ) , increases monotonically up to Emll = 4.10 kcal/mol with increasing collision energy. This increment of P(u = 1) is due to the energy transfer a t the deep well. In this deep well, the translational energy converts efficiently to the 0-H vibrational mode. The deep intermediate complex region causes the vibrational excitation of OH modes. At E,II = 5.3 1 kcal/mol, the OH(u = 1) fractional population decreases and deviates from the straight line. This feature is also in excellent accordance with the experimental Let us consider the reason why the OH(u = 1) fractional population decreases in this energy region (E,I, = 5.31 kcal/mol). As a working hypothesis, we consider here two mechanisms: (1) the OH(u = 2) product channel is open in this energy region, and (2) the excess energy of [OHFI- transfers to other internal modes (e.g. the rotational mode). Our theoretical calculation indicates that no OH(u = 2) product is found at Emll = 5.31 kcal/mol. Therefore, the former mechanism can be rejected. The ratio A / B calculated to be 0.35 at Eml,= 5.31 kcal/mol is slightly smaller than that of E,ll = 4.10 kcal/mol ( A / B = 0.43). This means that the preference of reaction channels is varied in this energy region. At collision energies larger than 4.10 kcal/mol, some of the trajectories, via direct channel, become nonreactive
+
The Journal of Physical Chemistry, Vol. 98, No. 20, 1994 5301
+
-
+
F + OH(u = 1, J = 0)
The product OH formed by the direct channel is in vibrational excited state (u = 1). The rotational-state population of the product OH peaks at J = 0 (Le., part A). The trajectory for this channel passes rapidly into the intermediate complex region. The other channel is a complex channel,
0-+ HF
-
F + OH(u = 0, J = highly excited)
The reaction for the complex channel proceeds via a long-lived intermediate complex [OHFI-. Therefore, the excess energy of [OHFI- formed by the collision can convert to the rotational energy within a lifetime. The rotational-state population of the product OH is widely distributed (i.e., part B). In a low collision energy region such as thermal energy (E,II = 1.198 kcal/mol), both channels are equivalently dominant. The product OH radical via the direct channel monotonically increases with increasing collision energy up to 4.10 kcal/mol. With a further increase in the collision energy (E,,, 5.3 1 kcal/mol), the product OH via the complex channel is dominant. Thus the present model explains the experimentally observed features of reaction I.6,7 C. Conclusion. The mechanism of proton-transfer reaction I is still in controversy, as described in section 1. In the present study, we propose a new reaction model for reaction I. This model is composed of two reaction channels (direct and complex channels). The model describes qualitatively the experimental features of the vibrational distribution deduced by Leone e?01.~9~ In the present calculation, we have introduced some approximations to construct the potential energy surfaces. In the PES calculations, we employ the U H F calculations with split valence plus polarization and diffuse function type 6-3 1++G** basis sets. Preliminary calculation at the D-CI+QC/6-3 1++G** level of theory gives essentially similar results for the shape of the PES.16 More elaborate calculations with a larger basis set and a more accurate wave function, such as MR-SD-CI or MCSCF-CIcalculations, are needed to obtain deeper insight into the collision process. Despite the approximations employed here, it is shown that a theoretical characterization of reaction I enables us to obtain valuable information on the mechanism of the collision process.
-
Acknowledgment. The authors are indebted to the Computer Center a t the Institute for Molecular Science (IMS) for the use of the computing facilities. H.T. acknowledges support from the Grant-in-Aid for Encouragement of Young Scientists (No. 5740344). References and Notes (1) (a) Baer, M. In The Theory of Chemical Reaction Dynamics; Baer, M., a s . ; Chemical Rubber: Boca Raton, FL, 1985; Vol. I!, Chapter 4. (b) Tachikawa, H.; Ohtake,A.; Yoshida, H. J . Phys. Chem. 1993,97,11944.(c) Tachikawa, H.; Hokari, N.; Yoshida, H. J . Phys. Chem. 1993,97,10035. (d) Tachikawa, H.; Murai, H.;Yoshida, H . J. Chem. Soc. Faraday Tram. 1993, 89,2369. (e) Tachikwa, H. Chem. Phys. Lett. 1993,212,27. (0Tachikawa, H.; Lunnel, S.;TBrnkvist, C.; Lund, A. Inr. J. Quanrum Chem. 1992,43,449; J . Mol. Srrucr (THEOCHEM) 1994,301,25. (g) Tachikawa, H.; Tomoda, S. Chem. Phys., io preu.
5302 The Journal of Physical Chemistry, Vol. 98, No. 20, 1994 (2) See, for example: State-Selected and State-to-State Ion-Molecule ReactionDynamics;Ng,C.Y.,Baer, M.,Eds.;AdvancesinChemicalPhysics; Wiley: New York, 1992; Part I, Experiment, Vol. 82, and Part 11, Theory. (3) (a) Varley, D. F.; Levandier, D. J.; Farrar, J. M. J . Chem. Phys. 1992, 96,8806. (b) Levandier, D. J.; Farley, D. F.; Farrar, J. M. J. Chem. Phys. 1992, 97, 4008. (c) Liao, C.-L.; Xu,R.; Flesch, G. D.; Baer, M.; N g , C. Y. J . Chem. Phys. 1990, 93, 4818. (4) (a) Squires, R. R.; Bierbaum, V. M.; Grabowski, J. J.; Depuy, C. H. J . Am. Chem. SOC.1983, 205, 5185. (b) Van Doren, J. M.; Barlow, S. C.; Depuy, C. E.; Bierbaum, V. M. Int. J. Spectrom. Ion Processes 1991, 109,
305.
(5) Langford, A. 0.;Bierbaum, V. M.; Leone, S. R. J. Chem. Phys. 1985, 83, 3913.
(6) Hamilton, C. E.; Duncan, M. A,; Zwier, T. S.; Weisshaar, J. C.; Ellison, G. B.; Bierbaum, V. M.; Leone, S. R. Chem. Phys. Lett. 1983, 94, 4. (7) Knutsen, K.; Bierbaum, V. M.; Leone, S. R. J . Chem. Phys. 1992, 96, 298. (8) Pulay, P. In Modern Theoretical Chemistry; Schaefer, H. F., Ed.; Plenum: New York, 1977; Vol. 4, Chapter 4. (9) (a) Hariharan, P. C.; Pople, J. A. Theor. Chim. Acta 1973,82,213.
(b) Francl, M. M.; Pietro, W. J.; Hehre, W. J.; Binkley, J. S.;Gordon, M. S.; DeFrees, D. J.; Pople, J. A. J. Chem. Phys. 1982, 77,3654. (c) Frish, M.
Tachikawa et al. J.; Binkley, J. S.; Schlegel, H. B.; Raghavachari, K.; Melius, C. F.; Martin, R. L.; Stewart, J. J. P.; Bobrowicz, F. W.; Rohlfing, C. M.; Kahn, L. R.; DeFrees, D. J.; Seeger, R.; Whiteside, R. A.; Fox, D. J.; Fleuder, E. M.; Topiol, S.; Pople, J. A. Ab initio molecular orbital calculation program GAUSSIAN86; Carnegie-Mellon Quantum Chemistry Publishing Unit: Pittsburgh, PA, 1986. (10) (a) Pople, J. A.; Binkley, J. S.; Seeger, R. Int. J . Quantum Chem. 1976, S10, 1. (b) Langhoff, S.R.; Davidson, E. R. Int. J . Quantum Chem. 1974,8,61. (c) Davidson, E. R.; Silver, D. W. Chem. Phys. Lett. 1977,52, 403. (d) Pople, J. A,; Seeger, R.; Krishnan, R. Int. J . Quantum Chem. 1977, S l l , 149. (1 1) (a) Meller, C.; Plesset, M. S. Phys. Rev.1934,46,618. (b) Bartlett, R. J. J. Phys. Chem. 1989.93, 1697. (c) Raghavachari, K. J . Chem. Phys. 1985,82, 4607. (12) Bunker, D. L. Methods Comput. Phys. 1971, 10,287. (1 3) Jonston, H. S. Gas Phase Reaction Rate Theory; Ronald: New York, 1966; Chapter 15. (14) Bradforth, D. W.; Arnold, D. W.; Metz, R. B.; Weaver,A.;Neumark, D. M. J . Phys. Chem. 1991, 95, 8066. (15) Heats of formation are cited from: Lias, S.G. J . Phys. Chem. Ref. Data 1988, 17, Suppl. 1. (16) Tachikawa, H. To be published.
