Accurate calculations of the rate constants and kinetic isotope effects

Accurate calculations of the rate constants and kinetic isotope effects for tritium-substituted analogs of the atomic hydrogen + molecular hydrogen re...
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J. Phys. Chem. 1083, 87, 3415-3419

3415

Accurate Calculations of the Rate Constants and Kinetic Isotope Effects for Tritium-Substituted Analogues of the H 4- H2 Reaction Donald 0. Truhlar;

Roger S. Grev,+

Department of Chemkby, Unhwslty of Minnesota, Minnsapdis, Minnesota 55455

and Bruce C. Garrett Chemical Dynamb Corporetion, t Columbus, ohlo 43220 (Received: October 18, 1982; In Final Form: March 2 1, 1983)

-

Rate constants are calculated for isotopic analogues of H + H2 H2 + H. The calculations are distinguished by (i) use of the most accurate potential energy surface available for any reaction, (ii) use of variationaltransition-statetheory, (iii) consistent inclusion of anharmonicity,and (iv) calculation of tunneling contributions by a three-dimensionalquantum-mechanicalanalogue of the collinear semiclassicalMarcus-Coltrin procedure. On the basis of previous comparisons of this kind of calculation to experimental data for five other isotopic analogues and to other theoretical calculations, we estimate the absolute accuracy of the calculations as approximately *50%, at least for 300-1000 K.

Introduction Transition-state theory provides a convenient framework for estimating rate constants of isotopic chemical reactions, but for most cases our a priori knowledge of the potential energy surface is insufficient for using the theory in a predictive way. Instead the theory is often used to obtain semiempirical potential energy surfaces or surface parameters for correlating rate data. Difficulties in making predictions even for given potential energy surfaces are that quantum-mechanical effects like tunneling are difficult to calculate and that a dividing surface through the saddle point does not provide an equally accurate transition state for all isotopic examples of a reaction.'s2 Nevertheless, we have shown that the best available ab initio potential energy surface3for H + H2 and dynamical calculations based on variational-transition-state theory with Marcus-Coltrin pathk' or small-curvature-tunnew3 vibrationally adiabatic transmission coefficients provide rate ~ o n s t a n t s ~ ~ ~in~ ~reasonable J"-'~ agreement with all available experimental data'&'* for the following isotopically analogous reactions: H + H2 Hz + H (R1) +

D

+ D2

+ H2 H + D2 MU + H2 D

D2 + D

(R2)

+H HD + D MuH + H

(R3)

+

4

-w +

DH

034)

(R5)

[Muonium (Mu) is the isotope of H in which the proton is replaced by a positive muon, with a mass about oneninth that of the proton.] We have also predicted rate constants MU + D2 MUD D 036)

-

+

There is no available experimental rate data for the other isotopic analogues of H H2but in this article we present predictions for the following reactions involving tritium: T + H2 H T + H 037)

+

T + D2

--

TD

+D

(R8)

t Lando Summer Undergraduate Research Fellow, 1979, 1980. Present address: Department of Chemistry, University of California, Berkeley, CA 94720. 1550 W.Henderson Rd.

*

0022-3654/83/2087-3415$01.50/0

These reactions are of course of interest in their own right, but the present results may be even more interesting for theoretical comparisons and for the general theory of kinetic isotope effects. The availability of these new rate constants means we now have reasonably accurate calculations for 14 isotopic analogues of a single reaction, and these can presumably be used for testing new theories or general relationships that may be proposed for treating kinetic isotope effects, especially tritium isotope effects, (1)B. C. Garrett and D. G. Truhlar, J. Am. Chem. SOC.,102, 2559 (1980);B. C. Garrett, D. G . Truhlar, and A. W. Magnuson, J. Chem. Phys., 76,2321 (1982);B. C. Garrett, D. G. Truhlar, A. F. Wagner, and T. H. Dunning, Jr., ibid., 78, 4400 (1983). (2)B. C. Garrett and D. G . Truhlar, J. Chem. Phys., 72,3460(1980). (3)B. Liu, J. Chem. Phys., 58,1925 (1973);P. Siegbahn and B. Liu, ibid., 68,2457(1978);D. G. Truhlar and C. J. Horowitz, ibid., 68, 2468 (1978);71,1514(E) (1979). (4)D. G. Truhlar and B. C. Garrett, Acc. Chem. Res., 13,440(1980). (5)R. A. Marcua and M. E. Coltrin, J.Chem. Phys., 67,2609 (1977). (6)B. C. Garrett and D. G. Truhlar, J.Phys. Chem., 83,200,3058(E) (1979). (7)B. C. Garrett and D. G. Truhlar, R o c . Natl. Acad. Sci. U.S.A.,76, 4755 (1979). ( 8 ) R.T. Skcdje, D. G. Truhlar, and B. C. Garrett, J. Phys. Chem., 85, 3019 (1981). (9)R.T. Skodje, D. G. Truhlar, and B. C. Garrett, J. Chem. Phys., 77, 5955 (1982). (10)B. C. Garrett, D. G. Truhlar, R. S. Grev, and A. W. Magnuson, J. Phys. Chem., 84,1730 (1980). (11)N. C. Blais, D. G. Truhlar, and B. C. Garrett, J.Phys. Chem., 85, 1094 (1981). (12)D. G. Truhlar, A. D. Isaacson, R. T. Skodje, and B. C. Garrett, J. Phys. Chem., 86,2252 (1982). (13)D. K. Bondi, D. C. Clary, J. N. L. Connor, B. C. Garrett, and D. G. Truhlar, J. Chem. Phys., 76,4986 (1982). (14)N. C. Blais, D. G. Truhlar. and B. C. Garrett, J . Chem. Phys., 76, 2768 (1982). (15)N.C. Blais. D. G . Truhlar. and B. C. Garrett. J. Chem. Phvs.. 78. 2363 (1983). (16)D. G. Truhlar and R. E. Wyatt, Annu. Reu. Phys. Chem., 27, 1 (1976). (17)G. Pratt and D. Rogers, J. Chem. SOC.,Faraday Tram. I , 72,1589 (1976). (18)D. G. Fleming and D. M. Garner, private communication. I

0 1983 American Chemical Society

.

.

3416

The Journal of Physical Chemistry, Vol. 87, No. 18, 1983

TABLE I: Rate Constants (em3molecule-' s - ' )

T,K

TST

CVT

Truhlar et al.

TABLE 11: Rate Constants ( e m 3 molecule-' s - ' )

ICVT/MCPVAG

T,K

TST

CVT

200 300 400 600 1000 1500 2400 4000

T + H,+TH+ H 4.88(-20)" 1.89(-20) 5.48(-17) 2.99(-17) 1.83(-15) 1.19(-15) 6.53(-14) 4.99(-14) 1.42( - 12) 1,18(- 12) 8.43(-12) 7.09(-12) 4.27(-11) 3.50(-11) 1.7 1(-10) 1.35(- 10)

1.01(-18) 1.94(-16) 3.74(-15) 9.12(-14) 1.58(- 12) 8.41(-12) 3.85(-11) 1.42( - 10)

200 300 400 600 1000 1500 2400 4000

H + T, 3.44( - 2 2) 1.7 9(- 18) 1.34(-16) 1.12(- 14) 4.93(-13) 4.08(- 12) 2.53(-11) 1.12(-10)

200 300 400 600 1000 1500 2400 4000

T+D,+TD+H 5.12(-21) 5.00(-21) 9.82(-18) 9.66(-18) 4.46(-16) 4.41(-16) 2.28(-14) 2.26(-14) 7.04(- 13) 6.97( - 13) 5.04(-12) 4.96(-12) 2.88(-11) 2.79(-11) 1.23(-10) 1.12(-10)

7.76(-20) 3.49(-17) 9.62(-16) 3.40(-14) 8.45(- 13) 5.55(-12) 2.97(-11) 1.16( - 10)

200 300 400 600 1000 1500 2400 4000

D + T, 1.07(-21) 3.35(-18) 1.97(-16) 1.33(- 14) 5.08(-13) 4.01(-12) 2.4 2( - 11) 1.06(- 10)

200 300 400 600 1000 1500 2400 4000

1.64(-21) 4.20(-18) 2.25(-16) 1.39(-14) 5.07(-13) 3.93(-12) 2.36(-11) 1.02(-10)

" In the tables, numbers in parentheses are powers of 10. on less well understood systems. In addition to our previous comparisons to experiment for reactions Rl-R.5, we have applied our theoretical model to purely collinear reactions for several isotopic analogues of H Hzand we have compared the predictions to accurate quantum dynamical calculations for several cases. Using the most accurate available potential energy surface, we have made such comparisons for reactions Rl,13R5,l2.l3 R6,1513and Rl2?J9 We have also made such studies using two earlier less accurate surfaces; in particular we have made comparisons for reactions R110J3920and R613 using the Porter-Karplus semiempirical valence-bond surface (their surface no. 2),2l and we have made comparisons for reactions R1,4WJo,l%mR2 6,8,10,1220 R3,8JOW and R41Otm for the rotated-Morse-curve surface of one of the authors and Kuppermann.22 The latter surface is a scaled fit to the ab initio calculations of Shavitt et al.,= which were earlier and less accurate than those reported in ref 3. The results of these collinear comparisons were generally very encouraging, with errors in the approximate calculations usually less than a factor of about 1.5. General discussions of variational-transition-state theory and the Marcus-Coltrin-path vibrationally adiabatic ground-state transmission coefficient are given else~ h e r e ~ Jand ~ , need ~ ~ not - ~ be ~ reviewed here. The discussion will therefore be restricted to the results for the reactions R7-Rl4 and to the kinetic isotope effects.

+

Theoretical Methods All calculations are carried out by using the ICVT/ MCPVAG version of variational-transition-state theory exactly as in ref 10. This is improved canonical variational (19)B. C. Garrett, D. G. Truhlar, R. S. Grev, and R. B. Walker, J. Chem. Phys., 73,235 (1980). (20)B. C.Garrett and D. G. Truhlar,J.Phys. Chem., 83,1079(1979); 84,682(E) (1980). (21)R. N. Porter and M. Karplus, J. Chem. Phys., 40,1105 (1964). (22)D.G. Truhlar and A. Kuppermann, J. Chem. Phys., 66, 2232 (1972). (23)I. Shavitt, R. M. Stevens, F. L. Minn, and M. Karplus, J. Chem. Phys., 48,2700 (1968). (24)D.G. Truhlar, J. Phys. Chem., 83,188 (1979). (25)B. C. Garrett, D. G. TRlhlar,and R. S. Grev in 'Potential Energy Surfaces and Dynamics Calculations",D. G. Truhlar, Ed., Plenum Press, New York, 1981,p 587. (26)D.G. Truhlar, A. D. Isaacson, and B. C. Garrett in 'The Theory of Chemical Reaction Dynamics",M. Baer, Ed., CRC Press, Boca Raton, FL, in press. (27)R. T. Skodje, D. G. Truhlar, and B. C. Garrett, J.Chem. Phys., 77,5955 (1982). (28)P.Pechukas, Annu. Rev. Phys. Chem., 32, 159 (1981). (29)P. Pechukas, Ber. Bunsenges. Phys. Chem., 86,372 (1982).

T

+ HT

+T

2.26( - 22) 1.37(- 18) 1.11(-16) 9.86(-15) 4.49(-13) 3.7 3(- 12) 2.29(-11) 9.84(-11) + DT

ICVT/MCPVAG 3.35(-21) 4.41(-18) 2.19(- 16) 1.38(- 14) 5.21(-13) 4.05( - 12) 2.39(-11) 1.01(-10)

+T

1.04(-21) 3.29(-18) 1.95(-16) 1.31(- 14) 5.04(-13) 3.96(-12) 2.3 7 (- 11) 1.01(- 10)

1.3 2(-20) 1.06(-17) 3.95(- 16) 1.8 9( - 14) 5.98(-13) 4.38(-12) 2.50( - 11) 1.04(- 10)

+ T,+T, + T 1.77(-20) 1.2 9( - 17) 4.47(-16) 2.00(-14) 6.04(- 13) 4.36(-12) 2.50(-11) 1.06(- 10)

(I

a u Q

a Q

u a

" Same as TST to number of significant figures tabulated. TABLE 111: Rate Constants (em3molecule-' s-I)

T, K

TST

CVT

ICVT/MCPVAG

200 300 400 600 1000 1500 2400 4000

T+HD+THtD 1.18(-20) 9.27(-21) 1.58(- 1 7 ) 1.38(- 17) 5.85(-16) 5.37(-16) 2.35(-14) 2.23(-14) 5.78(- 13) 5.39(-13) 3.7 1(-12) 3.30(- 12) 1.98(- 11) 1.65(- 11) 8.16(-11) 6.36(-11)

1.04(- 19) 4.17(-17) 1.05(-15) 3.15(-14) 6.34( - 13) 3.62( -12) 1.74( -1 1) 6.54(-11)

200 300 400 600 1000 1500 2400 4000

T+DH+TD+H 7.41(-21) 4.71(-21) 7.90(- 18) 1.06( - 1 7 ) 4.09(-16) 3.31(-16) 1.74(-14) 1.51(-14) 4.49( - 13) 4.06( - 13) 2.94(-12) 2.66(-12) 1.58(- 11) 1.4 1(- 11) 6.54(-11) 5.62(-11)

1.13(-19) 3.20(-17) 7.53(-16) 2.28(-14) 4.88(-13) 2.94( - 12) 1.49(-11) 5.7 9(- 11)

200 300 400 600 1000 1500 2400 4000

D+TH+DTtH 3.17(-21) 2.61(-21) 5.21(- 18) 5.90( -18) 2.63(-16) 2.40(-16) 1.30(- 14) 1.22( - 14) 3.77(-13) 3.62(-13) 2.60( - 12) 2.49( -12) 1.45(-11) 1.37(-11) 6.07(-11) 5.68(-11)

6.16(-20) 2.07(-17) 5.40(-16) 1.83(-14) 4.3 3(- 13) 2.76(- 12) 1.45(-11) 5.84(-11)

theory with a Marcus-Coltrin-path quantum-mechanical vibrationally adiabatic ground-state transmission coefficient. All calculations include Morse anharmonicity in the stretches (method IZo) and quartic anharmonicity in the bends,3Os3land rotations as well as vibrations are treated quantum mechanically.2J0The rate constant reported for reaction R11 is the distinguishable-atom one, but this is easily related to observables by straightforward methods.32 (30)B. C.Garrett and D. G. Truhlar, J. Phys. Chem., 83,1915(1979). (31)B. C.Garrett and D. G. Truhlar, J.Am. Chem. SOC.,101,4534 (1979).

Trltium-Substituted Analogues of the H

+ H, Reaction

TABLE IV: Intermolecular H/T Kinetic Isotope Effectsa T. K TST CVT ICVT/MCPVAG 200 300 400 600 1000 1500 2400

( H 4- H,)/(H + T,) 34.9 53.0 13.7 17.9 8.18 9.94 4.60 5.23 2.73 2.99 2.10 2.30 1.78 1.97

200 300 400 600 1000 1500 2400 4000

13.2 8.13 4.64 2.76 2.12 1.79 1.66

( D + H,)/[(D 200 300 400 600 1000 1500 2400 4000

3.66 2.74 2.32 1.90 1.55 1.39 1.28 1.24

9.93 6.70 4.17 2.58 1.97 1.63 1.47

34 2 44.2 17.1 6.88 3.37 2.47 2.04 92.7 20.6 10.4 5.20 2.83 2.06 1.65 1.47

+ HT) t ( D + T H ) ] 2.82 2.32 2.06 1.78 1.52 1.38 1.29 1.24

[ ( D t HT) t ( D + TH)l/(D t T,)

200 300 400 600 1000 1500 2400 4000

8.50 4.81 3.50 2.44 1.78 1.53 1.40 1.34

6.98 4.28 3.25 2.34 1.69 1.42 1.26 1.18

10.8 4.59 3.15 2.22 1.68 1.45 1.31 1.24 8.58 4.49 3.30 2.34 1.69 1.42 1.26 1.18

a Ratio of rate constant for top reaction to that for bottom reaction. Throughout this article (and in all our other papers on three-dimensional reactions), reaction rates for homonuclear (homoisotopic) diatoms refer to the sum of the reaction rates for reaction with both atoms of t h e molecule. For heteroisotopic diatoms, the atom transferred is listed first.

Results Tables 1-111 give the rate constants calculated for reactions R7-Rl4 by three methods: TST, conventional transition-state theory; CVT, canonical variational-transition-state theory2*20 with unit transmission coefficient, i.e., no tunneling; ICVT/MCPVAG, improved canonical variational theory’O with quantal effects on the reaction coordinate by the method used in ref 2,7, and 10. Tables IV-VI1 give various kinetic isotope effects as calculated a t the TST, CVT, and ICVT/MCPVAG levels of approximation. The convention of labeling kinetic isotope effects is explained in the footnote to Table IV. Discussion The kinetic isotope effecta (KIE’s)show several different kinds of trends. In every case we have tabulated them in the “normal” direction, i.e., lighter isotope in the numerator for intermolecular effects and lighter isotope transferred in the numerator for the intramolecular effects. The variational effects sometimes increase the KIE’s and sometimes decrease them. Tunneling usually, but not always (see Table VI), increases the KIE. The tables show how dangerous it is to make empirical generalizations about the variational effects (we use the term “variational effects”to denote those effects that c a w (32) D. G . Truhlar, J. Chem. Phys., 65, 1008 (1976).

The Journal of Physical Chemistty, Vol. 87, No. 18, 1983 3417

TABLE V: Intermolecular D/T Kinetic Isotope Effects T,K TST CVT ICVT/MCPVAG 200 300 400 600 1000 1500 2400 4000 200 300 400 600 1000 1500 2400 4000

( H + D J ( H + T,) 3.27 2.37 1.98 1.63 1.38 1.28 1.22 1.20

9.55 4.10 2.82 1.99 1.54 1.38 1.30 1.26

( D 4- H,)/(T t H,)

1.OB 1.09 1.10 1.10 1.10 1.10 1.10 1.10

0.683 0.806 0.874 0.942 0.990 1.01 1.02 1.02

( H + D,)/[(H 200 300 400 600 1000 1500 2400 4000

4.00 2.70 2.19 1.74 1.45 1.33 1.27 1.25

1.72 1.49 1.38 1.26 1.16 1.12 1.10 1.09

+

1.22 1.13 1.10 1.08 1.07 1.07 1.07 1.08

DT) t ( H + TD)] 1.93 1.61 1.45 1.31 1.20 1.15 1.12 1.11

[ ( H t DT) + H t TD)l/(H + T,

200 300 400 600 1000 1500 2400 4000

1.90 1.59 1.44 1.30 1.19 1.14 1.11 1.10

2.07 1.68 1.50 1.33 1.21 1.16 1.13 1.12

2.87 1.95 1.64 1.39 1.23 1.17 1.14 1.12 3.33 2.10 1.72 1.43 1.25 1.18 1.14 1.13

TABLE VI: Intermolecular H/D Kinetic Isotope Effects T, K TST CVT ICVT/MCPVAG 200 300 400 600 1000 1500 2400 4000

( T + H,)/(T + D,) 9.52 5.58 4.11 2.86 2.01 1.67 1.48 1.40

3.78 3.09 2.70 2.20 1.70 1.43 1.26 1.21

12.9 5.57 3.88 2.68 1.87 1.52 1.30 1.23

(T t H,)/[(T t HD) t ( T + DH)] 200 300 400 600 1000 1500 2400 4000

2.54 2.08 1.85 1.60 1.38 1.27 1.20 1.16

1.35 1.38 1.37 1.33 1.25 1.19 1.15 1.13

[(T + HD) t ( T t DH)l/(T + D,)

4.64 2.63 2.08 1.68 1.41 1.28 1.20 1.15

200 300 400 600 1000

3.74 2.69 2.23 1.79 1.46

2.80 2.25 1.97 1.66 1.35

2.79 2.12 1.87 1.60 1.33

1500

1.32

1.20

1.18

2400 4000

1.24 1.20

1.10 1.07

1.09 1.07

the CVT rate constants or KIE’s to differ from the TST ones). For example, one might have expected the (D + Hz)/[(D + HT) + (D + TH)] KIE’s to show the same variational effects as the (H + Dz)/[(H + DT) + (H + TD)] ones since in each case the attacked molecule Bzis

3418

The Journal of Physical Chemistry, Vol. 87, No. 18, 1983

TABLE VII: Intramolecular Kinetic Isotope Effects

T,K

TST

CVT

ICVTiMCPVAG

( T + HD)/(T + DH) 200 300 400 600 1000 1500 2400 4000

1.59 1.50 1.43 1.35 1.29 1.26 1.25 1.25

1.97 1.75 1.62 1.47 1.33 1.24 1.17 1.13

0.920 1.31 1.39 1.38 1.30 1.23 1.17 1.13

200 300 400 600 1000 1500 2400 4000

( D t HT)/(D t TH) 1.86 1.78 1.73 1.71 1.63 1.63 1.50 1.51 1.40 1.36 1.36 1.26 1.18 1.34 1.34 1.11

0.847 1.30 1.41 1.42 1.33 1.25 1.17 1.11

200 300 400 600 1000 1500 2400 4000

( H t DT)/(H + TD) 1.17 0.906 1.15 0.977 1.14 1.01 1.11 1.02 1.09 1.02 1.08 1.02 1.00 1.07 1.07 0.983

0.920 0.994 1.02 1.03 1.03 1.02 1.01 0.983

+

+

+

+

+

+

+

+

+

+

+

tests for the various isotopes, though, indicate that the ICVT/MCPVAG method is empirically reasonably accurate for all mass for this reaction. Tables 1-111 show that only one of the reactions studied here, T + T2,has its canonical variational transition state located at the saddle point. For the other reactions, the variational effects span the range of 2-370 (T + D2 and D + T,) to factors of 1.5-2.6 (H + T,, T + DH, and T H,) at 200 K, and they span the range 5-7% (D + T2and D + TH) to 27-2870 (T + H2and T HD) at 4000 K. We know from previous work that variational effects tend to be small for H + H2 and isotopic analogues compared to many other reactions; these small variational effects do, however, have very noticeable consequences for kinetic isotope effects, as we have discussed earlier in this section. The variational transition states for tritium-substituted reactions are compared to each other and to the H + H2 reaction in Table VI11 at temperatures of 300 and 2400 K. The quantities tabulated are the location s = S,~"~(Z') of the CVT transition state (the reaction coordinate is measured in the mass-scaled coordinate system of ref 6 and 20, with s < 0 on the reactants' side, s = 0 at the saddle point, and s > 0 on the products' side) and the following properties of CVT transition states [with * denoting that they are evaluated at s = S , ~ ( T ) ] RAB, : the A-B distance; RW, the B-C distance; Vm, the classical potential energy with respect to reactants; wStr,the harmonic frequency of the bound stretching mode; W p d , the harmonic frequency of the bending mode; and AVa ,the vibrationally adiabatic potential energy, including anharmonicity, with respect to reactants. Table VI11 shows several interesting features. When the temperature dependence of S . ~ ~ ( isT significant, ) the variational transition state always moves away from the saddle point as the temperature increases. The A-B and B-C internuclear distances for all the variational transition states in the table are in the range 1.53-2.05 ao, as compared to the saddle-point value of 1.76 ao. The classical potential energy at the variational transition state ranges as low as 8.76 kcal/mol (for T + HD at 2400 K), as compared to the saddle-point value of 9.80 kcal mol. Factor of the ratio k * ( T ) / k c ( T ) for the reactions studied in this paper show that, for T = 300 and 2400 K, the contribution of the rotational degrees of freedom is always in the range 0.94-1.00. Furthermore, at the same temperatures, the contribution of the bending degrees of freedom is always in the range 0.92-1.00, with the following exceptions: T + H2,0.81 at 300 K; T + HD, 0.75 at 300 K and 0.87 at 2400 K. Even for these reactions, however, the most significant effect determining the location of the variational transition state is the competition of the stretch and the potential energy along the reaction coordinate. Complete factor analyses are given in Table IX.

+

replaced by BT with mT > mB. However, the variational effects decrease the former KIE and increase the latter. A similar reversal is seen in comparing [(D + HT) + (D TH)]/(D T2) to [(H + DT) + (H + TD)]/(H + T2). Many of the KIE's show very large effects of tunneling, e.g., (H + H,)/(H + T2), (D + HJ/(D + TJ, (D + H,)/I(D HT) (D + TH)], and (H + D,)/(H + T,). This confirms previous analyses that showed that tunneling must be included in the transmission coefficients to explain the H + H, kinetic isotope e f f e ~ t s . ~ J ~ , ~ ~ v ~ ~ * ~ ~ The tables should provide a caution about conventional TST interpretations of KIE's without a more detailed analysis. See, e.g., the final KIE's for [(D + HT) + (D + TH)]/(D T,) or for (T H,)/(T D,). These KIE's agree very well with TST without tunneling even though both the variational effects and the tunneling effect show a significant isotope dependence. For another example, H,) or the [(T HD) (T see the (D H,)/(T DH)]/(T D2)ratio. If the ICVT/MCPVAG KIE in one of these cases had been an experimental one, and if it had been compared to TST for a reasonable potential energy surface, one might have concluded, in the absence of a variational calculation, that tunneling makes a large contribution to the KIE. Actually though the difference from TST is primarily determined by variational effects on the bottleneck location. In general the trends in the KIE's can be understood only by understanding the variational effects and tunneling corrections for each of the individual reactions involved. Tunneling effects are most important at low temperature, of course. For the reactions studied here, quantum effects are most important for T + H,, T + DH, and D + TH. Notice, in particular, that the tunneling correction is not always largest for the transfer of the lightest atoms. It is probable that the present calculations are more accurate for T + DH, say, than for T + HD, and that they underestimate the tunneling for T + HD.'JO Our collinear

+

Truhlar et al.

+

+

(33) D. G. Truhlar, A. Kuppermann, and J. T. Adams, J. Chem. Phys., 59, 395 (1973). (34) D. G. Truhlar, A. Kuppermann, and J. Dwyer, Mol. Phys., 33,683 (1977).

A

Comparison to Previous Work The best previous transition-state calculations of the tritium kinetic isotope effects are those of S h a ~ i t t . ~ ~ Shavitt used conventional transition-state theory, the harmonic approximation, a conservation-of-vibrationalenergy tunneling model, and a scaled version of the potential energy surface of ref 23. The present calculations are distinguished from previous attempts to calculate these rate constants by (i) use of the most accurate potential energy surface available for any chemical r e a ~ t i o n , ~(ii) ,~' (35) B. C. Garrett and D. G . Truhlar, J . Am. Chem. SOC.,101, 5207 (1979). (36) I. Shavitt, J . Chem. Phys., 49, 4048 (1968).

Tritium-Substituted Analogues of the H

+ H,

The Journal of Physical Chemistry, Vol. 87,No. 18, 1983 3410

Reaction

TABLE VIII: Variational Transition States A

+ BC

H+H,

T+H,

T+D,

0.00 1.76 1.76 9.80 2059 910 9.26

-0.21 2.01 1.55 8.93 2646 815 8.94

-0.03 1.78 1.73 9.78 1344 626 9.16

0.00 1.76 1.76 9.80 2059 910 9.26

-0.20 2.00 1.55 8.98 2601 819 8.94

-0.16 1.94 1.60 9.30 1628 606 9.04

H+T,

D+T,

T+T,

T+HD

T+DH

D+TH

0.03 1.73 1.78 9.78 1343 548 9.7 1

0.00 1.76 1.76 9.80 1190 526 9.45

-0.21 2.04 1.53 8.80 2231 759 8.87

-0.12 1.89 1.63 9.51 2003 676 8.99

-0.10 1.85 1.67 9.65 1932 626 9.19

0.12 1.66 1.86 9.60 1448 54 2 9.67

0.00 1.76 1.76 9.80 1190 526 9.45

-0.21 2.05 1.53 8.76 2265 756 8.87

-0.18 1.96 1.59 9.22 2231 664 8.99

-0.12 1.87 1.65 9.57 1992 623 9.18

T = 300 K 0.17 1.64 1.88 9.56 1944 608 10.53

T = 2400 K S*CVT, a , R * A B , a0 R*BCv V*MEp, kcal/mol w *str/2nc, cm-I W

*bad/2nC, c m - '

AVaG(s*CVT), kcal/mol

TABLE IX:

0.20 1.63 1.90 9.48 2008 606 10.53

proximation is lower than that for D + H2.In the CVT approximation the D H2rate constant exceeds the H H, one at 600 K, and both the D + H2and T + H2rate constants exceed the H H, one at T = 200-400 K.

Factors in k T S T (T)/kCVT( T)

+ BC

potential

T + H, T + D, H + T, D + T, T t T, T + HD T + DH D + TH

0.233 0.977 0.674 0.977 1.00 0.188 0.613 0.787

T + H, T + D, H + T, D + T, T + T, T + HD T + DH D + TH

0.842 0.901 0.935 0.960 1.00 0.805 0.885 0.954

A

stretch

T = 300 10.3 1.04 1.99 1.04 1.00', 8.51 2.28 1.47

rota tion

kTST/

0.940 0.998 0.964 0.998 1.00 0.956 0.966 0.980

1.83 1.02 1.31 1.02 1.00 1.15 1.34 1.13

0.943 0.973 0.955 0.985

1.22 1.03 1.11 1.02

1.00

1.00

0.954 0.941 0.972

1.20 1.12 1.05

bend

+

kCVT

K 0.814 1.00 1.01 1.00

1.00 0.749 0.990 1.00

T = 2400 K 1.66 1.24 1.24 1.10 1.00 1.79 1.39 1.14

0.925 0.948 1.00 0.986 1.00 0.873 0.968 0.997

use of variational-transition-state theory, (iii) inclusion of anharmonicity, (iv) use of the three-dimensional analogue of the Marcus-Coltrin tunneling path, which is particularly successful for H + H,and isotopic analogues. In particular, as discussed elsewhere, we believe that the vibrationally adiabatic type models are more consistent than conservation-of-vibrational-energy models with transition-state theory, and they are empirically much more accurate.103XW9 The only other results to which we can compare are the quasiclassical trajectory calculations of Mayne@for reactions R7 and R9. Mayne calculated reaction cross sections as functions of relative translational energy for the ground vibrational-rotational state. The trends in the cross sections were (T + H,)> (D + H,)> (H+ H,)> (H+ D2) > (H + T,). In the present work we never find the thermal rate constant for T H2to be larger than that for D H2, at either the CVT or ICVT/MCPVAG levels of approximation. At high temperature (T 1 1000 K) we find the thermal rate constants to be ordered as (H+ H,)> (D H,)> (T H,)> (H D2) > (H Tz). At 200-600 K the H + H2rate constant in the ICVT/MCPVAG ap-

+

+

+

+

+

+

+

factors

+

(37)D.G.Truhlar and R. E. Wyatt, Adv. Chem. Phys., 36,141(1977). (38)D.G.Truhlar and A. Kuppermann, J.Am. Chem. SOC.,93,1840 (1971). (39)D.G.Truhlar and A. Kuppermann, Chem. Phys. Lett., 9, 269 (1971). (40)H.R.Mayne, J. Chem. Phys., 73,217 (1980).

Summary The H + H2reaction and its isotopic analogues are the only ones for which we have an accurate enough potential energy surface3l3' available to calculate ab initio rate constants at room temperature to an accuracy comparable to that available from current experiment techniques. It has been demonstrated previously that canonical or improved canonical variational theory with Marcus-Coltrin-path vibrationally adiabatic ground-state transmission coefficients (ICVT/MCPVAG) and this potential energy surface can account within experimental accuracy for the available experimental results on the H + H,and D + D2ortho-para H + D,, and Mu + conversion reactions and the D + H2, Hzexchange r e a c t i ~ n s . ~ ~ ' We J ~ Jhave ~ used the accurate potential energy surface to perform ICVT/MCPVAG calculations for eight isotopic analogues involving tritium. Since these rate constants are also believed to be of high accuracy, they may be useful for applications or testing simpler models, and they are reported here for reference purposes. The methods are identical with those explained elsehere.^^^^^^^^^^^^^^^^^ In particular all results include anharmonicity in the stretches by Morse model I and in the bends by a quadratic-quartic model. All partition functions and one-dimensional transmission probabilities are calculated fully quantum mechanically (no classical or semiclassical approximations). The rate constants are tabulated in Tables 1-111, and some of the kinetic isotope effects are tabulated in Tables IV-VII. Both the rate constants and the kinetic isotope effects are compared to the predictions of conventional transition-state theory with unit transmission coefficient and canonical variational theory with classical reaction-coordinate motion, Le., with unit transmission coefficient. We also summarize the locations and properties of the variational transition state and present a factor analysis of the variational effects. Acknowledgment. The work at the University of Minnesota was supported in part by the U.S. Department of Energy, Office of Basic Energy Sciences, under contract no. DE-AC02-79ER10425. The work at Chemical Dynamics Corp. was supported by the US. Army through the Army Research Office under contract no. DAAG-28-81-C0015 Registry No. H, 12385-13-6; Hz, 1333-74-0;D2,7782-39-0; Tz, 10028-17-8.