J. Phys. Chem. 1986, 90, 1152-1 155
1152
The current set of results should provide a demanding test of modern statistical theories of liquid mixtures. Perturbation theory was successfully used by Clancy, Gubbins, and Gray15 to predict the excess Gibbs energies and the excess volumes of xenon ethene at the triple point temperature of xenon. This theory has also proved successful for ethane ethene rnixturesl6 but failed for xenon ethane.2 This seems to indicate that, in order to simultaneously account for the properties of the xenon + ethene and the xenon + ethane systems, a perturbation theory will have to use a nonspherical reference system. A promising scheme is
Acknowledgment. This work was supported by Grant No. CPE-8 104708 from the National Science Foundation. M.N.P. thanks the Council for International Exchange of Scholars and JNICT-INVOTAN (Portugal) for grants. Registry No. Xe, 7440-63-3; ethene, 74-85-1.
(15) Clancy, P.; Gubbins, K. E.; Gray, C. J. Discuss. Faraday SOC.1978, 66, 116. (16) Calado, J. C. G.; Azevedo, E. J. S. G.; Clancy, P.; Gubbins, K. E. J . Chem. Soc., Faraday Trans. 1 1983, 79, 2657.
( 1 7) Azevedo, E. J. S.G.; Lobo, L. Q.; Staveley, L. A. K.; Clancy. P. Fluid Phase Equilib. 1982, 9, 267. ( 1 8 ) Fischer, J.; Lago, S.; Lustig, S . ; Bohn, M. to be submitted for publication.
+
+
+
the perturbated hard-sphere treatment of Azevedo et a].” A more elaborate procedure that gave good results for xenon ethane at the lower temperatures is the Boltzmann averaging perturbation theory of Fisher et al.’*
+
CHEMICAL KINETICS Thermodynamic and Kinetic Properties of the Cyclohexadienyl Radical W. Tsang Chemical Kinetics Division, National Bureau of Standards, Gaithersburg, Maryland 20899 (Received: January 28, 1985; In Final Form: October 7, 1985)
The experimental data on cyclohexadienyl (1,3-cyclohexadien-5-y1)radical decomposition have been examined. In combination with an estimated entropy for cyclohexadienyl of 375 f 7 J/(K mol) at 550 K, and the measured rates of hydrogen addition to benzene and the reverse we find the heat of formation of cyclohexadienyl to be 209 f 5 kJ/mol ( 3 0 0 K j . This leads to exp(-13100 & 700/7‘)/s in the vicinity of 550 K. The HC-H bond energy k(cyclohexadieny1- benzene + H) = 1013,3”0,6 in cyclohexadiene is 318 kJ, leading to a resonance energy of 96 kJ. The second-order disappearance rate of cyclohexadienyl radical at 425 K has been found to be 7 X lo-” cm3/(molecule s). This is in excellent agreement with direct measurements at room temperature.
Introduction
Recently, Nicovich and Ravishankaral published the results of a study on the reactions of hydrogen atoms with benzene. From the temperature region where hydrogen atom decay was nonexponential they found the rate expression for cyclohexadienyl (1,3-cyclohexadien-5-y1 and related forms) decomposition to be
-
k(cyclohexadieny1
benzene + H ) = 1.3 X 10l6 exp(-16700 f 3900/T)/s
In combination with the rate expression for the reverse reaction k(benzene H cyclohexadienyl) = 6.7 X lo-’’ exp(-2170/p cm3/(molecule s)
+
they obtained equilibrium constants whose temperature dependence was such that a heat of formation for cyclohexadienyl radical of 191.2 f 11.7 kJ/mol (298 K) was derived. This is in satisfactory agreement with the recommendation of McMillen and Golden2 for a heat of formation for cyclohexadienyl of 196.6 f 2 1 kJ/mol. Earlier, Benson and co-workers3 recommended values ( 1 ) Nicovich, J. M.; Ravishankara, A. R. J . Phys. Chem. 1984, 88, 2534. (2) McMillen, D. F.; Golden, D. M. In “Annual Reviews of Physical Chemistry”; Rabinovitch, B. S.. Ed.; Annual Reviews: Palo Alto, CA, 1982; 493. (3) (a) Benson, S. W. “Thermochemical Kinetics”, 2nd Ed.; Wiley: New York, 1976. (b) Shaw, R.; Cruickshank, F. R.; Benson, S. W. J . Phys. Chem. 1961, 71. 4538.
in the 184-209 kJ/mol range. The experimental results of James and Suart4 yield
-
-
+
k(cyclohexadieny1 benzene H) /k’’2(2cyclohexadienyl combination and disproportionation products) = exp(-15710 f 2370/T) (molecules/(cm3 S))I/~ Combining this with the results of Sauer and Ward5 leads to a decomposition rate of 10*6.0 exp(-15710/T)/s. On this basis they recommend a heat of formation of cyclohexadienyl of 184 kJ/mol. These are very difficult experiments. Although the similarities in rate expressions may lead to the conclusion that the results support each other, examination of Figure 1 shows that there are gross disagreements in extrapolated absolute rates if we take the rate expressions at face value. However, the experimental rate expressions are highly uncertain. Note that the 32-kJ uncertainty in the activation energy from the results of Nicovich and Ravishankaral implies an uncertainty in the A factor of a factor of 1000. Of interest is the extraordinarily high A factor for unimolecular decomposition. This would suggest “loose” complexes and are much different from the “normal” complexes for alkyl radical decomposition.6 In this note we present an alternative treatment of the data. We begin by showing the consequences (4) James, D. G. L.; Suart, R. D. Trans. Faraday Soc. 1968, 64, 2752.
(5) Sauer, M. C.; Ward, B. J . Phys. Chem. 1967, 71, 3971. (6) O’Neal, H. E.; Benson, S.W . In “Free Radicals”; Kochi, J. K., Ed.; Wiley: New York, 1973; Vol. 2, p 272.
This article not subject to U S . Copyright. Published 1986 by the American Chemical Society
The Journal of Physical Chemistry, Vol. 90, No. 6, 1986 1153
Properties of the Cyclohexadienyl Radical 42
I
I
I
I
TABLE I: Frequencies and Contributions to Entropy at 550 K for Benzene and 1,4-Cyclohexadiene
2
freq, cm-’ (degeneracy),entropy, J / ( K mol) benzene 1,4-~yclohexadiene (a) Ring Deformation 106, 19.1; 250, 12.0; 405, 410 (2), 16.24; 606 (2), 10.62; 703, 8.2; 530, 6.23; 572, 5.70; 4.35; 1010, 2.30 897; 2.97 XS = 54.2 = 33.5
cs
\
3 h , \
(b) Ring Stretches 854, 3.18; 956, 2.55; 1377, 992, 2.40; 1310, 1.20; 1486 (2), 1.71; 1.09; 1405, 1.09; 1639, 1596 (2), 1.34 0.61 1680, 0.60, = 9.0 XS = 6.65
cs
(c) C-H Bends 613, 4.32; 849 (2), 6.44; 975 (2) 4.94; 622, 5.15; 706, 4.31; 995, 2.43, 1000, 2.34; 1035, 995, 2.38; 1038 (2), 4.35; 1150, 1.72; 2.18; 1159, 1.67; 1193, 1178 (2), 3.3; 1362, 1.18 1.59; 1187, 1.55
oc
ES = 28.6 18
20
22
24
(d) CH2 motions
900, 2.87; 956, 2.57; 1100
1000/ T
(2), 3.81; 1358 (2), 2.22,
Figure 1. Summary of experimental (-) and calculated (-- -) results on cyclohexadienyl decomposition: A, data of Nicovich and Ravishankara; B, data of James and Suart. The envelope about each line represents the uncertainties in rate constants as given in their text by these investigators.
with respect to the structure of cyclohexadienyl radical if the thermochemistry of Nicovich and Ravishankara is correct. Specifically, this involves comparison of the derived entropy with that of benzene. We then try to make an estimate using benzene and 1,4-~yclohexadieneproperties as representative of the two extreme limits. Combination with the hydrogen addition data of Nicovich and Ravishankara leads to a more reliable value for the heat of formation of cyclohexadienyl radical and a new and less uncertain rate expression for its decomposition. Finally, as a check of the utility of the procedure we show that we can reproduce the directly measured rate of cyclohexadienyl radical c~mbination.~
Procedures Nicovich and Ravishankara’ derived their heat of formation from the relation d In K,,/d(l/T) = -AH/RT This “second-law” method is self-contained but is dependent on an accurate measurement of the slope. Small errors in the measured equilibrium constant can easily result in large discrepancies in the reaction enthalpy. Alternatively, the heat of the reaction can be determined through the “third-law” method via the relation AH = TAS - R T In Keq For this purpose it is necessary to make an estimate of the entropy for the reaction. If this can be done, then the uncertainty is measured equilibrium constant will be reflected in the heat of reaction through the logarithm. For example, at 550 K a factor of 2 uncertainty in equilibrium constant will lead to an uncertainty of 3.2 kJ in the enthalpy. Since the entropies of H atoms and benzene are well-known, the key to this procedure is the estimation of the entropy of cyclohexadienyl. The entropies of many hydrocarbon radicals can be estimated with high precision by using standard statistical mechanical formula^.^.' The rotational constants are little changed in comparison with appropriate hydrogenated or dehydrogenated species. At reasonable temperatures, only a few of the vibrational frequencies make important contributions to the entropy. These can be estimated from comparable systems with a high degree of accuracy. It is significant that the entropies of ethyl and n-propyl radicals were accurately (7) Tsang, W. J . Am. Chem. SOC.1985, 107, 2812.
1 s = 21.2
1426, 0.97; 1430, 0.96
XS =
13.4
(e) C-H stretches c S = 0.15
X S = 0.27
determined from estimated vibrational frequencies close to 20 years before the vibrational frequencies were actually measured.* We have recently employed this procedure to rationalize the data on alkane decomposition and radical combination9 and alkyl radical decomposition and radical addition to olefins.’ These have led to new values for the heat of formation of alkyl radicals and the demonstration that rejection of decomposition rates in the past has been due to an unwarranted attachment to low enthalpic values. In the following we make use of Nicovich and Ravishankara’s data at 550 K (midpoint of their experiments) and determine AHf(cyclohexadienyl) on the basis of the estimated A S of reaction. An expression for the equilibium constant is determined and, in combination with the rate constant for the reverse addition process, the rate expression for decomposition is calculated. Applying this expression to the results of James and Suart leads to the rate constant for the interaction of two cyclohexadienyl radicals.
Analysis A correct heat of formation will be compatible with secondand third-law treatments. We can test Nicovich and Ravishankara’s heat of reaction of 11 1.7 kJ by calculating the entropy of reaction dictated by their measured equilibrium constant for the reaction H + benzene 2 cyclohexadienyl At 550 K, they find the equilibrium constant to be 2 X 104/atm. This leads to A S = -120.9 J / ( K mol). Since S(C6H6) = 339.3 J / ( K mol)loaand S(H) = 127.2 J/(K mol),Iob then S(C6H7)= 345.6 J/(K mol). The latter is very close to that for benzene. However, the symmetry factor for benzene is 12. Therefore, the intrinsic entropy for benzene at 550 K is 360 J/(K mol). For the cyclohexadienyl radical, the symmetry number is 2 , but the electronic degeneracy is 2. Thus, the intrinsic entropy remains 345.6 J/(K mol). A necessary consequence of their results is that cyclohexadienyl radical possess a “tighter” structure than benzene. (8) Pacansky, J.; Horne, D. E.; Gardini, G. P.; Bargon, J. J . Phys. Chem. 1977, 81, 23. (9) Tsang, W . Int. J . Chem. Kinet. 1978, 10, 821. (!O) (a) Stull, D.; Westrum, E. F.; Sinke, G. C. “Chemical Thermodynamics of Organic Compounds”; Wiley: New York, 1967. (b) Stull, D. R.; Prophet, H. “JANAF Thermochemical Tables”; U.S. Government Printing Office: Washington, DC; NSRDS-NBS-37.
1154
The Journal of Physical Chemistry, Vol. 90, No. 6, 1986
The possibility of this occurring can be assessed from a comparison of the vibrational frequencies of benzene” and 1,4-cyclohexadiene.I2 These are summarized in Table I. At 550 K, C-H stretches contribute little to the entropy. The extra H atom added to benzene has the effect of converting the two frequencies associated with a C-H bend to the four frequencies associated with two CH2 motions. For our purposes, we have selected average values (920 and 1 150 cm-I for the C H bends and 1400 and 1000 cm-I for the CH, motions). This leads to an entropy increase of 1.59 J / ( K mol). This new value, 361.6 J / ( K mol), can now be considered the intrinsic entropy of the cyclohexadienyl radical based on benzene structure. In order to reduce this value to the experimental number of 345.6 cal/(K mol) considerable tightening of the benzene structure is required. For example, if the lower frequencies of the ring deformation modes of cyclohexadienyl were to be raised to 850 cm-’, then the entropy deficit can be made up. This is a highly unlikely possibility. One notes that 1,4cyclohexadiene has in fact a much looser structure than benzene. We can also start from the other direction and compute the entropy of cyclohexadienyl radical using the frequencies of 1,4cyclohexadiene as a base. The vibrational frequency assignment and structural parameters of Stidham12lead to an intrinsic entropy of 388.7 J / ( K mol). If we now replace the four frequencies associated with the C H 2 nonstretching motions with the two frequencies associated with the C-H bend we arrive at an intrinsic entropy of cyclohexadienyl radical of 387.1 J/(K mol). The entropy of cyclohexadienyl must lie between the two extremes represented by the benzene (361.6 J / ( K mol)) and 1,4-cyclohexadienyl structures (387.1 J / ( K mol)). We can reduce this spread somewhat by noting that the structure of the resonance stabilized cyclohexadienyl radical resembles 1,4-~yclohexadiene at the end that contains the CH, group while the opposite end is essentially benzene-like. One also notes that for a compound such as cy~lopentene’~ the ring puckering motion at 550 K has an entropy very similar to that for cyclohexadiene. On this basis we assign ring puckering motions for cyclohexadienyl radical of 106 and 410 cm-I. The corrected entropies based on benzene and 1,4 cyclohexadiene structures are then 368.2 and 382.4 J/(K mol), respectively. Since these two numbers represent extremes, we arrive at an intrinsic entropy of 375.3 f 7.1 J/(K mol). This is an average of the two numbers. The entropy correction from the rotational symmetry and the electronic degeneracy cancel; thus the entropy of the radical remains 375.3 f 7.1 J/(K mol). This is equivalent to a heat of reaction of 95.4 f 5 (550 K). Note that we could have also used 1,3-~yclohexadieneas a starting point. The end result would have been the same since the vibrational frequency assignments and hence the intrinsic entropies of the two isomers aree strikingly ~ i m i l a r . ’ ~ This * ’ ~ is hardly surprising. We have chosen to work with 1,4-cyclohexadiene because the vibrational assignments seem to be better established and it is more convenient for our subsequent purposes. With these values we can now compute thermodynamic properties of cyclohexadienyl radical as a function of temperature. This is done by making small changes in the frequency assignments within the framework described earlier so as to obtain an entropy of 375.3 J / ( K mol) at 550 K. No claim is made that these are the actual frequencies of cyclohexadienyl. Thermodynamic functions are not very sensitive to the specific vibrational assignment of frequencies. The entropy of 375.3 J/(K mol) at 550 K represents a calibration point and our assignments should be good enough to reproduce the temperature dependence over the 300-1 000 K range. Thermodynamic properties are summarized in Table 11. This leads to a heat of formation of cyclohexadienyl ( 1 1 ) Shimanouchi, T. Natl. Stand. Ref. Data Ser. (U.S., Natl. Bur. Stand.) 1972, No. 39. (12) (a) Stidham, H. D. Spectrocbem. Acta 1965, 21, 2 3 . (b) Laane, J.; Lord, R. C. J . Mol. Spectrosc. 1971, 39, 346. (13) Laane, J.; Lord, R. C. J . Chem. Phys. 1967, 47, 4941. (14) Carreira, L. A.; Carter, R. 0.;Durig, J. R. J. Chem. Phys. 1973, 59, 812. (15) Carreira, L. A,; Lord, R. C.; Malloy, T. B. Top. Curr. Chem. 1979, 82. 1.
Tsang TABLE 11: Calculated Thermodynamic Properties of Cyclohexadienyl Radical“ temp C., J/K S, J/(K mol) Hi, kJ/mol 200 298.16 300 400 500 600 700
273.27 301.3 301.83 331.29 360.93 389.79 417.39 443.58 468.36 491.77
57.24 86.37 86.98 118.97 146.94 169.68 188.35 203.79 216.72 221.64
800 900
1000
215.64 208.90 208.80 202.70 197.83 193.98 190.99 188.75 187.14 186.17
+
“log Kr = -12.7137 - 8967.621T - 4 0 5 8 2 6 / p 2.2368 X 107/T3; lxl,lz= 7.64 X gm3 cm6. Frequencies: ring deformations: 106, 410, 630, 640, 650, 1035; ring stretches: 920, 956, 1358, 1620, 1650; C-H bends: 700, 706, 920, 950, 1100, 1130, 1150, 1159, 1197; CH2 motions: 950, 950, 1420, 1450; CH stretches: 2822, 2875, 2877, 3077, 3019, 3032, 3032. (The frequencies given here are estimates that yield the desired entropy value a t 550 K. They are based on assignment for benzene and 1,4-cyclohexadiene and the pattern should be sufficient to permit extrapolation to 298 K.)
of 208.9 kJ/mol (298 K). When this is coupled with Nicovich and Ravishankara’s rate expression and uncertainties for the addition process we arrive at the rate expression k(cyclohexadieny1 benzene + H ) = 1013.3*0.6 exp(-13100 f 700/7‘) s-’
-
It should be emphasized that unlike experimental studies this activation energy is not the result of a direct slope measurement. Instead, it is based on the heat of reaction at the appropriate temperature and the activation energy of the reverse process. The uncertanity in the rate constant is the same as that given by Nicovich and Ravishankara for their decomposition measurements or a precision of 15% (at 550 K) since we use their number as a base. It will increase somewhat as we go to different temperatures. As a check of this result, we find at 425 K, near the midpoint benzene of the work of James and Suart, k(cyclohexadieny1 H ) = 0.75/s. Using their value for k(cyclohexadieny1 benzene H)/k1I2(2cyclohexadienyl combination and disproportionation products) of 8.9 X lo4 (molecules/(cm3 s)’I2 we find k(2cyclohexadienyl combination + disproportionation products) = 7 X IO-” cm3/(molecule s). This is in extraordinarily good agreement with the direct measurements of Sauer and Ward which yielded 8.3 X lo-’’ cm3/(molecule s). Although Sauer and Ward expressed surprise at their high value, this is due to their erroneous surmise that the combination of resonance-stabilized radicals should be much slower than that for alkyl radicals. We now know that there is very litle difference. Thus Tulloch et a1.I6 find for allyl radicals k , = 2.7 X lo-’’ cm3/(molecule s).
-
+
+
-
-
- -
Discussion On the basis of the present third-law treatment all the existing rate constant rate on the H benzene cyclohexadienyl reaction are in quantitative agreement providing that the proper thermochemistry is used. In contrast, had we simply extrapolated the rate expression of Nicovich and Ravishankara to 425 K we would have obtained k(2cyclohexadienyl combination and disproportionation products) = 1.5 X lo-’* cm3/molecule s), or a factor of 50 smaller than the directly measured number. Similarly, if we begin with the results of James and Suart and Sauer and Ward and extrapolate to 550 K we would have obtained rate constants a factor of 4 higher than the measured value. The above is illustrative of the important difference between the results of Nicovich and Ravishankara and James and Suart. The fact that the rate expression look similar and only differ by 8.3 kJ in activation energy is illusory since the variations in activation energies have large effects in the rate constants in the temperature
+
-
(16) Tulloch, J. M.; MacPherson, T. M.; Morgan, C. A,; Pilling, M . J. J . Phys. Chem. 1982, 86, 3812.
J . Phys. Chem. 1986, 90, 1155-1160 TABLE III: Rate Expressions for H-Atom Ejection from Some Hydrocarbon Radicals
no. of
----
k(iC3H7 k(rC4H, k(sC4H, k(sC4H9 k(~CqH9 k(cC6H7
available H rate expressions
C3H6+ H) = exp(-18704/T)/s C4H8 + H) = exp(-18910/T)/s C4H8-1 + H) = exp(-l8498/T)/s cC4H8-2+ H) = exp(-l7650/T)/s tC4H8-2 + H) = I O i 2 ’ exp(-17225/T)/s C6H6+ H) = exp(-l3090/T)/s
atoms 6 9 3 1 1 2
range of concern. Our approach, in which we give first preference to the measured rate data, represents the “standard”l7 procedure for looking a t kinetic results. It is quite reasonable since even small errors in rate determinations can have large consequences in activation energies. This then makes extrapolations from such results extremely uncertain. The general situation is summarized in Figure 1. Our rate expression for cyclohexadienyl decomposition is completely different from that of Nicovich and Ravishankara and James and Suart (using Sauer and Ward’s numbers on cyclohexadienyl disappearance at lower temperatures). Nevertheless, as can be seen in Figure 1, the discrepancies in rate constants are close to or within the stated error limits. We therefore conclude that our thermodynamics and rate expression resolves the discrepancies between the rate measurements. In a larger sense, this exercise demonstrates the value of the third-law approach to the treatment of equilibrium data. It is interesting to compare the present results with similar data for H-atom ejection from alkyl radicals. The are summarized (17) Benson. S.W.: ONeal. H. E. “Kinetic Data on Gas Phase UnimoIeculaiReaction’s”;U.S: Government Printing Office: Washington, DC; 1970 NSRDS-NBS 21.
1155
in Table 111. The rate expression for the latter have been derived in the same manner as this study. Within experimental error, and taking into account the reaction path degeneracy, the A factor for cyclohexadienyl decomposition is very close to that for alkyl radicals. Apparently, resonance stabilization and cyclization have very small effects. The 209 kJ/mol heat of formation is higher than that derived by Nicovich and Ravishankara and suggested by McMillen and Golden. It leads to a HC-H bond dissociation energy of 1,4cyclohexadiene of 3 18 kJ. Associating this with our secondary C-H bond strength of 414 kJ7 we arrive at a resonance energy of 96 kJ. It should be noted that earlierI8 we derived a C-H bond strength for cyclohexane of 414 kJ from the decomposition of tert-butyl cyclohexane. From our heat of formation of 174.5 kJ/molI9 for allyl radical and 100.8 kJ/mol for n-propyl radical we find the allylic resonance energy to be 50.2 kJ. Thus the cyclohexadienyl radical possesses close to twice the allylic resonance energy. This is very similar to values proposed in the past. However, these were based on significantly lower C-H bond energies. The new values for cyclohexadiene HC-H bond proposed here and for the secondary C-H bond proposed earlier have the effect of cancelling each other. In the case of cyclopentadienyl and 1,bpentadienyl radicals the resonance energy appears to be of the order of 75 W.This is in line with theoretical predictions.20 Such theories will predict the same value for cyclohexadienyl. There is thus a rather large discrepancy. Acknowledgment. This work was supported by the Department of Energy, Division of Chemical Sciences, Office of Basic Energy Science under Interagency Agreement DE-A-101-76PR06010. Registry No. 1,3-Cyclohexadien-5-yl,15819-51-9. (18) Tsang, W. J . Phys. Chem. 1972, 76, 143. (19) Tsang, W. J . Phys. Chem. 1984,88, 2812. (20) Herndon, W. C. J . Org. Chem. 1981, 46, 2119.
Theoretical Analysis of Kinetic Isotope Effects in Radical Reactions. Deuterium Isotope Effects in Inter- and Intramolecular Additions. Dependence on Temperature J. Igual and J. M. Poblet* Department of Quimica- Fisica, Facultat de Qurmica de Tarragona, Universitat de Barcelona, Pc. Imperial Tarraco s/n. Tarragona 43005, Spain (Received: February 22, 1985; In Final Form: July 10, 1985)
A MIND0/3 theoretical study within the framework of the transition-state theory has been realized on the secondary and primary kinetic isotope effect (KIE) in different radical reactions. The dependence on temperature has been analyzed by partitioning the total isotope effect into its rotational, translational, vibrational excitation, and zero-point contributions. A comparison of KIE between the inter- and intramolecular additions has been performed and the role of the new vibrations is discussed.
Introduction The elucidation of organic reaction mechanisms has been realized extensively using the kinetic isotope effects’ (KIE). However, the arguments used in the theoretical interpretation of observed isotope effects have been of a very qualitative nature, and many of the now accepted explanations are open to criticism. One of the major problems lies in the arbitrary approximation made in order to estimate the vibrational frequencies associated with the assumed transition structures. Now, the M I N D 0 / 3 , 2 ( 1 ) van Hook, W. A. In “Isotope Effects in Chemical Reactions”; Collins, C. J., Bowman, N. S., Eds.; Van Nostrand: New York, 1970.
0022-3654/86/2090-1155$01.50/0
MND0,3 and ab initio calculations can provide all the necessary information. In the past years the M I N D 0 / 3 and M N D O methods have been extensively tested with regard to the calculation of kinetic isotope effect^.^-^ Ab initio calculations have also (2) Bingham, R. C.; Dewar, M. J. S.; Lo, D. H. J . Am. Chem. Sor. 1975, 97, 1286. (3) Dewar, M. J. S.; Thiel, W. J . Am. Chem. Sor. 1977, 99, 4899. (4) Brown, S. B.; Dewar, M.J. S.; Ford, G. P.; Nelson, D. J.; Rzepa, H. S. J . Am. Chem. SOC.1978, 100, 7832. (5) Dewar, M. J . S.; Ford, G. P. J . A m . Chem. Sor. 1977, 99, 8343. Dewar, M. J. S.; Olivella, S.; Rzepa, H. S. J . Am. Chem. Sor. 1978, 100, 5650. (6) Rzepa, H . S. J . Chem. Sor., Chem. Commun. 1981, 939.
0 1986 American Chemical Society
