JOURNAL OF THE AMERICAN CHEMICAL SOCIETY (Registered in U. S. Patent Office)
(0Copyright, 1961, by the American Chemical Society)
JANUARY 16, 1961
VOLUME 83
NUMBER 1
PHYSICAL AND INORGANIC CHEMISTRY [CONTRIBUTION FROM THE
DEPARTMENT OF
CHEMISTRY, UNIVERSITY O F CALIFORNIA,
BERKELEY4,
CALIFORNIA]
Large Tunnelling Corrections in Chemical Reaction Rates.I I1 BY HAROLD S. JOHNSTON
AND
DONALD RAPP
RECEIVED APRIL 26, 1960 In terms of certain lengths near the saddlepoint of an activated complex relative to the de Broglie wave length of the atom being transferred, the reaction coordinate of bimolecular atom-transfer reactions is profitably classified as: (1)essentially classical, (2) essentially separable or (3) non-separable. In terms of the location of the saddlepoint and energy-distance curvatures through the saddlepoint, simple general rate expressions for case 1 and case 2 are given, including a small degree of tunnelling, utilizing the recent general method (ref. 6) of evaluating the configuration integral of any molecule. Hydrogen atom transfer reactions below several hundred degrees centigrade are in the region of non-separable reaction coordinates. For these cases more information about the potential energy surface is needed than the saddlepoint geometry and curvatures. Sample calculations using the Sato-potential energy surface for Ha, as evaluated by Weston, illustrate an approximate method for treating non-separable reactions, including large degrees of non-separable tunnelling. The hydrogendeuterium isotope effect in reactions of methyl radicals with hydrocarbons is worked out in detail, and this method of handling large degrees of tunnelling appears to agree with the experimental data over a wide temperature range (aithough there is large experimental error).
Introduction
stant potential energy, the “classical reaction path” AOB, and with several straight lines drawn in. The potential energy4 along the ‘‘classical reaction path” is given by AOB in Fig. 2, and an extended parabola based on the curvature a t 0 is given by
I n a previous article2 referred to as T-I we pointed out the anomalies in the rates of hydrogenatom abstraction reactions by methyl radicals. The data clearly required some tunnelling correction, and yet such a correction based on a one-dimensional Eckart potential along the classical reaction path greatly overestimated the rate. The comparison of H3C-H-D and H3C-D-H also indicated these reactions have about equal tunnelling factors with tunnelling occurring along modes of motion different from the normal modes of small vibration theory. We developed the thesis that the usual procedure of treating the reaction coordinate as a one-dimensional, separable coordinate cannot be employed if the de Broglie wave length A* for the atom being transferred is large compared to the linear portions (both in the RI-RZ plane and along the potential energy profile) of the potential energy surface around the saddlepoint. The meaning of these restrictions is well brought out by considering the classical reaction path in the R1-R2 plane, AOB in Fig. 1, and by considering the energy profile along this path, AOB in Fig. 2 . Fig. 1 shows several features of a conventional potential energy surface3 with contour lines of con-
POQ. It is convenient to recognize three special cases, although in fact the three situations flow continuously from one to another. (1) T h e reaction coordinate i s essentially classical: The criterion is that the de Broglie wave length A* of the atom being transferred is short compared to the essentially linear portion of AOB near 0 in Fig. 1 and is short compared to the essentially flat-topped portion of AOB near 0 in Fig. 2 . (2) T h e reaction coordinate i s separable but not classical: I n an area near the saddlepoint 0 in Fig. 1 the changes in potential energy on the surface may be adequately approximated by 2 A V = F,,(ARi)*
+ F2z(ARz)’ + 2FizARiAR2
(1)
where A’s refer to displacements from the saddlcpoint. The criterion for the separability of the reaction coordinate is that each dimension of this (3) S. Glasstone, K. J. Laidler and H. Eyring, “Theory of Rate Processes,” McGraw-Hill Book Co.. New York, N. Y.,1941. (4) T h e features of Fig. 2 are t o scale and are based on Dr. Ralph Weston’s Sat0 potential for the Ha activated complex, R. E. Weston, J. Chem. PhTs., 31, 892 (1959).
(1) In part from Ph.D. thesis by D. Rapp, Univ. of Calif., 1959. (2) H. S. Johnston, Advances in Chemical Physics, 3 , in press (1960).
1
HAROLD S. JOHNSTON
3
\
1.
Fig. 1.-Portion of a potential energy contour map for atom transfer reaction, where the activated complex is symmetrical with respect to bond lengths RI and Rt and to force constants along these bonds. AOB is the “reaction path”; COD is the extended symmetric stretch normal mode; EOF is the extended reaction coordinate or reaction normal mode; GH is a line parallel t o EOF a t a “shorter crossing distance”; and LM is a line parallel t o EOF a t a “longer crossing distance.”
area be long compared to A*; that is, the dimensions of the quadratic region near the saddlepoint 0 must be large compared to A*. The region over which the parabola POQ in Fig. 2 corresponds adequately to the reaction path AOB may be used to estimate the extent of the quadratic region; i t is about i 0.3 A.for Ha. (3) The reaction codrdilzate i s non-separable: The criterion is simply that the de Broglie wave length A* is larger than the dimensions of the quadratic region. The de Broglie wave , a hydrogen ?tom a t length, h ( 2 ~ m h T ) - ‘ / ~for 300’K. is 1.01 A. and a t 500’R. is 0.78 A. Thus reactions involving hydrogen transfer a t temperatures within several hundred degrees of room temperature fall into the third category! This article develops an approximate method for treating special cases of reactions which fall under group 3. Further and more fundamental work on this topic is underway.6 Classical Reaction Coordinate.-For a flat topped barrier (that is, flat relative to A*) i t is convenient to consider two sub-classifications : (a) all internal vibrations are also classical and (b) the other internal vibrations are quantized. For the reaction Ax B = A xB, the completely classical problem can be set up with no conceptual difficulties. If reactants Ax and B have an equilibrium distribution over their excited molecular states, then properties of the activated complex with respect to arrival of reactants should be averaged over equilibrium distributions. If an imaginary line is placed through and normal to the saddlepoint (point 0 in Fig. 2), the flux of systems through this line from the side of reactants is
+
AND
DONALD RAPP
Vol. 83
Fig. 2.-Potential energy profiles along various lines in Fig. 1. Energy and distance are to scale for the H activated complex according t o Weston’s calculations of a Sat0 potential. The line POQ is the extended parabola that has the same curvature -F*as EOF a t 0. In fitting a n Eckart function t o curve EOF, the parameters are the negative curvature F* and the energy difference AV*.
where QA=and QB are partition functions for reactants] Q+ is the complete partition function (3N dimensional) for the activated complex, Q* is the phase integral for the reaction coordinate] ( l / h ) J : 6 ; 2 smme-H*lkT dqdp, and theother terms areconventional. Eyring,a reasonably enough, prefers to express both Q+ and Q* in normal mode coordinates, so that Q* cancels the identical term in Q* leaving Q*‘ (3)
where K is the probability that the particle go on to products after it has once crossed the line above the saddlepoint. An alternate method, which is ultimately identical though quite different in the computational stages of the problem, is to express Q+ / Q A ~ Q Bin terms of local valence-bond coordinates,6 to express Q* in terms of the reaction coordinate normal mode and to derive (appendix 1) the rate expression for a single reaction site and single electronic state (4)
The configuration integrals are given in the recently-derived,6 general, simple form
+
(6) R. S.Pitzer and E. M. Mortensen, private communication.
where iV1 is the number of atoms in the reactant Ax, N is the number of atoms in the activated complex] IF, / A = is the determinant of the force constant matrix for Ax in local valence bond coordinates, IFs\ is the force constant matrix based directly on the various curvatures (as demonstrated in
*
(6) D. R . Herschbach, H.S . Johnston and D. Rapp, J . Chem. P h y s .
si, 11362(ia59).
Jan. 5, 1961
3
LARGE TUNNELLING CORRECTIONS IN' REACTION RATES
dent packet of particles with a Boltzmann distribution of energies for the two potential energy barriers shown in Fig. 3. The solid line is a symmetrical Eckarta potential of height TI* and curvature F* a t the maximum
-
V(r)
=
V*
(8)
coshz [x( F*/2 V*)%]
The dotted line in Fig. 3 is a truncated parabola with the same barrier height, V*, and curvature, - F*, as the Eckart potential. V(X)
*V
- '/,F*X~
(9)
The Schroedinger equation for the Eckart potential can be solved, and for a n incident particle of mass p and energy E the transmission coefficient is
Fig. 3.-Symmetrical Eckart potential energy function and the truncated parabola with the same negative curvature at the maximum, F*, and the same total height V*. The plot of potential energy against distance is expressed in dimensionless form.
4
where
=
E/V*, a = 2nV*/hv* and v* =
If 4a2>> r 2 and for f = 1, the transmission function for the Eckart potential approaches the WKB solution for the parabolic barrier, which islo K(€) = [1 4- exp d l - i11-l (11) As a approaches infinity one obtains high flattopped barriers and consequently the transmission functions for both parabolic and Eckart barriers approach the classical form (F*/p)'/'.
appendix 2) through the saddlepoint of the activated complex. (An expression similar to 5A applies to the other reactant.) I n (4)v* is the imaginary "frequency of the reaction coordinate," and i t is given by (1/2 T ) ( X * ) ' / ~ where A* is the negative K(5) = 0, 5 < 1; K(E) = 1, 5 > 1 root of IFG - EA1 = 0 for the activated complex.? (12) A general method for finding J= and a convenient The form of the transmission functions for the paratable for typical molecular groupings are given in bolic barrier for various values of a relative to a ref. 6. classical barrier is given in Fig. 4; similar curves for I n case b, as defined above, that is, real vibrations Eckart barriers are given in Fig. 5 . Numerical of the reactants are quantized, we must include in the denominator a correction factor FLAX= (ui/2)/sinh (ur/2) =
Qsu/QEi
(6)
where ui = hvi/kT. For the vibrations of the activated complex orthogonal to the classical reaction normal mode, we may also add a factor I'i* of the same form as (6). However, here we encounter a conceptual difficulty; as pointed out by Kasse18the use of quantum mechanical stationarystate partition functions for a species as transient as an activated complex is highly questionable. For the moment, bypassing this difficulty, the rate expression for classical reaction coordinate and quantized orthog onal vibrations is 3N
-7
.
< -
t
_ iI *
( = r
-
I
1
' 2
c2
+
Fig. 4.-Transmission
coefficients of a parabolic barrier as a function of the energy of an incident particle for various values of the parameter 01 = 2?rV*/hv*. For quantum mechanical systems, one notes reflections above the barrier as well as penetration beneath the barrier.
where the I"s all have the form of eq. 6 . One Dimensional Barrier Penetration.-In this section we wish to extend the work of KemblelS Bell,l0 Eckart, l1 Shavitt12and others3by presenting the numerical and graphical results of the transmission coefficient as a function of energy and the distribution of transmitted systems from an inci(7) E. B . Wilson, Jr., J. C. Decius and P. Cross, "Molecular Vibrations," McGraw-Hill Book Co., New York, N. Y.,1955. (8) L. S. Kassel, J . Chem. Phyr.. S, 399 (1935). (9) E. C. Kemble, "Fundamental Prinaples of Quantum Mechanics," McGraw-Hill Book Co., New Ybrk, N . Y.,1937. (10) R . P. Bell, Trans. Faraday Soc., 66, 1 (1959). (11) C. Eckart. P h w . Rev.. 86. 1303 11930). (12) I. Shavitt; J . Chrm. P h y s l , 31, 1359 (1959).
L
"
8
,
3
c 3
2
I? < = .
4
" 2,
L ?
f
Fig. 5.-Transmission coefficients of an Eckart barrier as a function of the energy __ of an incident particle for various values of the parameter a = 2aV*/hv*.
4
HAROLD
s. JOHNSTON
;IND
VOl. 83
DONALD RAPP
R ,
“
212, L
$
I
L
L
-
_
_
:rI;\
2 , then the parabolic barrier has the simple tunnelling correction functionlo
r* =
(u*/2)/sin
( % * / a ) ;u* < ?T;
Q!
>2
(13)
Finally, i t is necessary in many cases to consider how the tunnelling factor I?* changes with temperature. The parameter e* is defined as O* = T d In T*/dT
(14)
For the parabolic barrier correction as given by (13) this function is [(u*/2)cot (u*/2) - 11. Separable Reaction Coordinate.-Insofar as the primary making and breaking of a bond is concerned, an atom transfer reaction usually is discussed in terms of a potential energy contour plot
Jan. 5, 1961
LARGE
TUNNELLING CORRECTIOXS I N REACTION RA4TES
5
TABLE I TRANSMISSION COEFFICIENTK(&) FOR SYMMETRICAL ECKART BARRIERSAS A FUNCTION OF INCIDENT ENERGY, 5 = E / V*, AND FOR DIFFERENT VALUESOF CY = 2 r V * / h I u * l e E 0.1 .2 .3
.4 .5 .6
.7 .8 .9
1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
20
0.070 .070 .0,1 .0646 .Ob92
.0116 .016 .127 .531 .889 .981 .997 .999 1,000 1.000 1.ooo 1,000 1,000
1.000
8
12
16
0.072 .0660 ,0691 ,0499 .0386 ,0262 .038 ,184 .539 ,848 .961 .990 ,998 ,999 1,000 1.000 1,000 1.000 1.000 1.000
r*
0.079 .os21 .Oa24 ,0818
.0*11 .0255 .024 .089 .264 .551 .799 ,924 .973 .990 ,996 .998 .999 1.000 1.000 1.000
TUNNELLING FACTORS, = kqu/kI1, u*
1.5 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 12.0 14.0 16 0
a+
20
1.1 1.2 1.5 2.0 3.2 6.11 14.9 46 199
. ...
16
1.1
1.2 1.5 2.0 3.21 5.69 12.3 32 107 437
....
....
.... ....
....
....
....
....
....
0.0&24 .0320 ,0398 .0238 .012 .036 .090 ,201 .375 .577 ,749 .863 ,928 ,962 .980 .990 ,994 .997 .998 .999
FOR 12
1.1 1.2 1.5 2.0 3.07 5.21 9.90 22 55 162
4
3
0.0267
0.028 .070 .129 .201 ,287 .379 ,471 ,558 ,637 .705 ,762 .809 ,847 .878 .902 .922 .937 .949 .959 .966
.021 .047 .090 .153 ,238 .339 .449 ,557 .655 .737 ,803 ,853 .892 .920 ,941 .956 .967 .975 .981
1
2
0.115 .229 .335 .432 ,517 .591 .654 ,708 .752 .790 .821 .848 ,870 .889 ,904 .918 ,929 .938 ,946 .953
0.50
0.25
0.791 .884 .920 .939 .951 .959 ,965 .970 .973 .976 .978 .980 .982 ,983 .984 .986 .987 .987 .988 ,989
0.455 ,633 .728 ,787 .826 .855
,877 .893 .906 ,918 .927 .934 ,941 ,946 ,951 .956 .959 ,962 .965 ,968
0.940 ,969 .979 .984 .987 .990 .991 .992 ,993 .994 .994 .995 .995 .996 ,996 .996 ,996 .997 .997 .997
TABLE I1 SYMMETRICAL ECKART POTENTIALS AS A FUNCTION OF u* = h
Iu*~
0.125
0.984 .992 .995 .996 .997 .997 ,998 ,998 .998 ,998 .999 .999 .999 .999 .999 ,999 ,999 .999 .999 .999
/ k T AND
CY
8
4
3
2
1
0.5
0.25
0.125
1.16 1.24 1.56 2.04 2.94 4.54 7.60 13.8 27 57 307
1.18 1.30 1.58 2.02 2.69 3.69 5.23 7.60 11.3 17.3 42 110 304
1.20 1.31 1.60 2.00 2.57 3.38 4.52 6.15 8.48 11.9 24 50 107
1.22 1.32 1.57 1.91 2.34 2.90 3.62 4.55 5.76 7.34 12.1 20 34
1.19 1.27 1.43 1.61 1.83 2.09 2.38 2.72 3.11 3.56 4.68 6.19 8.19
1.12 1.16 1.25 1.34 1.44 1.55 1.67 1.80 1.94 2.09 2.42 2.81 3.25
1.06 1.08 1.12 1.17 1.22 1.26 1.31 1.37 1.42 1.47 1.60 1.73 1.87
1.03 1.04 1.06 1.08 1.10 1.13 1.15 1.17 1.20 1.22 1.27 1.32 1.37
... . ....
on a RI-R~diagram such as Fig. 1. On this diagram there are three extremes along the “classical reaction path,” infinitely separated reactants, infinitely separated products, and the activated complex near the saddlepoint of the potential energy surface. I n a Taylor series expansion about each of these three points, the three constant terms define: an origin of energy, the activation energy a t absolute zero and the heat of reaction a t absolute zero, not including zero point energy. The first derivative of each is zero by virtue of its being an extremum ; and for sufficiently small displacements about each point the potential energy surface is adequately described by quadratic terms, as in eq. 1. It is only within the quadratic region that the vibrations are separable normal modes. For somewhat larger displacements the normal mode coordinates are recognizable and convenient. and they are used with corrections for the perturbing effect of “anharmonicity” and for slight coupling with respect to energy transfer. For even larger displacements the normal coordinates lose their identity and usefulness; the system in general becomes one of non-separable, interacting coordi-
nates. The reaction coordinate is LLseparable’’ over the region near the saddlepoint where the potential energy surface may be regarded as quadratic as in eq. 1. The rate expression for the separable parabolic reaction coordinate involves a tunnelling factor for the one dimensional normal mode
r* =
Kq, ( E )R E ) dE/Jum Rei ( E ) P ( E ) dE
(15)
where the integral in the numerator is the area under the dotted curve in Fig. 6 or 7 and the denominator is the area under the classical curves on the same figures. The rate expression for the separable reaction coordinate is thus Rate (separable) = R(eq. 7) r* (eq. 13) (16a)
This equation can be expressed in the form
u
Rate
=
(classical, eq. 4)
r3N-af
(16b)
arreaotants
I n this way we see one of the advantages of expressing the rate in terms of the classical configuration integral with quantum corrections; the tun-
6
HAROLD S. JOHNSTON
AND
DONALD RAPP
VOl. 83
nelling factor for the separable reaction coordinate as a quasi-one dimensional symmetrical Eckartappears naturally as one of the 3N - 6 quantum potential tunnelling problem. factors for the activated complex. It is an amusing Dr. Weston evaluated in detail a potential energy coincidence that the quantum correction factors surface for the H-H-H r e a ~ t i o n . ~He has kindly have the same form, (u/2)/sinh(u/2), where u is let us borrow his detailed numerical values of the real for real vibrations, and for the reaction coordi- potential energy function. We are using these data nate i t is imaginary, convertingsinh(zd/2) tosin(u/2). to explore the effect of large scale tunnelling. The Another comparison of eq. 3 and 4 will be pointed saddlepoint for Hs is a t 0.93-0.93A. The reaction out. When use is made of eq. 3, one goes to great normal mode is a straight line of -45' slope pains to evaluate moments of inertia for the mole- through 0.93-0.93, EOF of Fig. 1. The energy cule as a whole or for internal rotors, if they are profile of such a straight line is given to scale in present. However, these moments of inertia al- Fig. 2 as EOF. A parabola with the same curvaways identically cancel mass terms in the compli- ture a t 0.93-0.93 is included in Fig. 2 , and we see cated expressions for the product of vibrational fre- that the quadrFtic region along this energy proiile is quencies in the classical limit of the vibrational about f 0.3 A , ; the de Broglie owwavelength of a partition function. Thus eq. 4 is equivalent to eq. hydrogen atom a t 500'K. is 0.78 A. The extended 3, but with eq. 4 one never needs to compute mo- reaction coordinate EOF encounters side-wall rements of inertia and other terms which cancel. A pulsion and eventually turns upward. The minicommon approximation in using eq. 3 is to treat mum energy is 1.4 kcal. below the saddlepoint, vibrational partition functions as unity. If some AV* of Fig. 2, while the energy of reactants is 8.6 vibrational frequencies are indeed low, then in kcal. below the saddlepoint. Parallel cuts (-45') effect one puts the moment of inertia in the numer- a t different crossing points, 0.96-0.96 (LM), 0.90ator but fails to put the compensating terms in the 0.90 (GH) are shown in Figs. 1 and 2 . For crossing low frequency vibrational partition function in the points snorter than the saddlepoint, GH, the curvadenominator. This approximation leads to sub- ture is less sharp (smaller F*), the energy from crossstantial errors, up to a factor of lo*, in polyatomic ing point to minimum is less deep (smaller AV*) activated complexes. and the crossing point is higher than a t the saddleNon-separable Reaction Co6rdinate.-If the de point; all of these effects tend to decrease the tunBroglie wave length associated with the atom being nelling for short crossings. For crossing points transferred is large compared to the quadratic longer than the saddlepoint, Lhl, F* is greater and portion of the potential energy surface, then the A V* is greater than a t the saddlepoint; these effects reaction coordinate is not separable from the real tend to give more tunnelling than a t the saddlevibrations. In this section we wish to explore a point. However, the height of the barrier increases simple, though inexact, method for handling this as one moves away from the saddlepoint, and this situation. e % ' express the rate as effect via Boltzmann's factor tends to counteract the more favorable trend in F" and A V*. Rate = A series of cuts a t -45: was made for crossing points varying every 0.01 A. from 0.80, 0.80; 0.81, 0.81 . . . 0.93, 0.93; . . . 1.09, 1.09. For each such cut a value of F and A V was found and a t each of where Kqu(a,E) is the quantum mechanical trans- three temperatures, 333, 500 and 1000°K., p* and mission coefficient of a particle with energy E LY were evaluated for H-H-H and D-D-D. For (relative to reactants) and coordinates q arriving each such cut the Eckart transmission coefficient from reactants, K,l(q,E) is the classical transmission from Table I1 is interpreted as the value for K(g), coefficient (zero or one) and P ( E ) is the probability where g is the crossing point for a -45' motion. The plot of K(q) against g is given in Figs. 9, 10, 11 of energy E in the coordinates q. When A and B are heavy atoms or groups and x for H-H-H and D-D-D a t each of three temperais H or D, we have the simplest case with which to tures. The form of the classical curve is given by test (17). Normal mode analysis (with FA^* = the Roltzmann factor for the crossing point, and F x*)~ gives a reaction coordinate in which dRA, = all transmissions are normalized to unity for classi- d R X ~that , is, RABremains constant and x moves cal transmission just a t the saddlepoint. The from X i o B with a small motion of A-B to conserve area under the classical curve is the denominator center of mass. The reduced mass of this motion is of eq. 17, and the area under the quantum curves essentially mx/2 (the factor of 2 arises because a is the numerator of eq. 17. Thus the relative ~ subtracts areas give an estimate of a two-dimensional tunsingle unit of motion of x adds to R Aand from R x to~ give a double motion in dRAx - dR,B) nelling correction, r*non-ssep. Using Figs. 9-11 we can evaluate tunnelling This coijrdinate corresponds to motion through the Any other angle correction factors by the several different methods saddlepoint with a slope of -43'. of crossing corresponds to forced motions of the and compare the results in Table 111. The ratios heavy end groups, A and R , and to a much higher of quantum to classical areas under the curves in effective mass. Thus the integration in (17) can Figs. 9-11 are entered as I'*av. If we take these to be carried out for different values of the crossing be the best estimate of tunnelling, we see that a t point, but in all cases for a -45' crossing angle, as low temperatures a parabolic correction greatly between GH and LM of Fig. 1. The "approxi- overestimates tunnelling, and a one dimensional mate method" mentioned in the introduction is that Eckart correctiori based on the reaction path (AOB, the line through each such crossing p o h t is treated in Fig. 1) also greatly overestimates tunnelling.
L ~ R GTUSh'ET21.1NC E CORRECTIONS IN REACTION RATES
Jan. 5, l9Gl
c
1
pcl~'dTLl
_J
, &%=LU-Ll .--
0.53
Figs. 9-11.-Relative rates of barrier crossing for various lines between HG and LM in Fig. 1. Each value of K ( g ) corresponds t o the area under a curve such as Figs. 6-8. a reaction and the DI Calculations are made for the H reaction. Relative areas under these curves give the values of r*sy.in Table 111. Relative values a t the saddlepoint Relative values a t the maxima give r*=.*. give I'*r.o.
slightly. At low temperatures the Boltzmann factor sharpens the classical distribution of systems which react and tunnelling becomes of much greater TABLE 11: COMPARISON OF TWO-DIMESSIONAL TUNNELLING FACTORS WITH VARIOUS ONE-DIMESSIONAL TUNSELLING FACTORS Reaction
H-H-H
Corrections a b c
d 6
CF:Sj,NG
"C
-.
D-D-D
0
b
A one dimensional Eckart correction based on the normal mode coordinate (EOF in Fig. 1) fairly seriously underestimates the tunnelling correction. One fortunate simplification emerges ; the values of K ( q ) for the most probable crossing point on CD in Fig. 1 that give the maximum in Figs. 9-11 are in rather good agreement with the tunnelling factor based on the averages. (This feature is not uncommon : the average is well approximated by the most probable situation.) The carrying out of the computations to construct Figs. 9-11 is very tedious. The constructive simplification discovered by this analysis and used to interpret experimental data is to replace the averages in (17) by most probable values, to give the practical, working re1a t'ion Rate (non-sep) = R(eq. 7 ) P (2 dim., most prob.) (18)
Other aspects of Figs. 9-11 are worthy of comment. At 1000°K., both classically and in terms of quantum mechanics, the systems which react were widely distributed in crossing points and the classical and quantum distributions differ only
c
d
Tunnelling factors r* = k,,*/k,i* 533'K. 500°K. 1000'K.
5.84 5.31 3.85
2.66 2.54 2.20 3.75
1.41 1.41 1.38 1.40
...
6.4 1.80 1.78 1.70 1.9
1.40 1.19 1.18 1.1: 1.18
lG.6
3.20 3.13 2.70 4.6
2.02 1.18 1.48 1.19 b 1.44 1.20 E 1.30 1.18 d 1.97 1.18 e 3.16 1.18 ... tunnelling factor averaged over various crossing points, two dimensional average, the area under quantum curve divided by area under classical curve in Figs. 9, 10,ll. I'*m.p.,the value of r*(q)a t the most probable crossing point, the maximum value of K(q) in Figs. 9, 10, 11. "I'*r.o.,the value of K(q) along the reaction coordinate normal mode through the saddlepoint; the value of K(q) a t 0.93 in Figs. 9, 10, 11; note that 4 V * in the Eckart relation is 1.3 kcal., giving a low value of a. r*r.p., the value of K(q) along the "classical reaction path" extending from reactants t o complex to products. The value of A V* is the full A V*,,t or 7.8 kcal. The Eckart parameter CY is thus much larger than for (c), although the P,p , Y*, or u* are all the same. The difference between (c) and (d) is due to the importance of "side wall repulsions" to extended normal mode tunnelBell's parabolic relation, eq. 13. ling. e
H-H-H D-D-D
a
12.8 1..82 1.70 1.43 3.60
S
HAROLD
101
4 m 2000 1000 1
I
T,
O K .
l
500
333
F;
a I--
s. JOHNSTON ._
/]
AND
DOXALD RApp
3
I ~
2c
10
30
lO00/ T. Fig. 12.-Calculated and observed kinetic isotope effect for methyl radical reactions with organic compounds, (-4) Calculated curve assuming no tunnel effect, where the ' z zero kinetic isotope effect arises largely from ( m o / m ~ ) ~and point energy differences of reactants. ( B ) Calculated curve assuming tunnel effect based on rXrnas found from treatments similar to but less extensive than Figs. 9-11. (C) Calculated curve assuming Eckart potential along AOB of Fig. 1 and CY given by 2 ~ V * ~ ~ t j h u * .
relative importance. However, a t low temperatures the Boltzmann factor largely wins out over the tunnelling phenomenon with respect to locating the reaction sites. The most probable classical crossing point is 0.93-0.93, and a t 333'K. the most probable quantum reaction trajectory is through 0.95-0.95. Even a t the lowest temperatures, where the tunnelling factor is almost 6, the distribution function for quantum systems which cross the symmetric stretch axis is surprisingly similar to that for classical systems. The nature of the London-Eyring-Sat0 potential energy surface is such that the reaction process is much more nearly along the classical normal mode than one would expect, simply from a comparison of the de Broglie wave length and the region of separability. Methyl Radical-Hydrocarbon Kinetic Isotope Effects.-For the family of reactions
+
XBC J-CR
where x is H or D and y is H or D, a portion of the Sat0 potential energy surface was evaluated. The activation energy (corrected for zero point energy) was taken to be 13.6 kcal.2 The activated complex was assumed to have RI = Rz, and the saddlepoint was estimated from Pauling's rule of bond orderbond distance to be near p = A R = 0.18 A. For
83
various values of Sato's parameter k between 0.10 and 0.20 and for values of p between 0.15 and 0.25, a relatively small number of potential energy points were evaluated, and the correct saddlepoint was quickly found to be given by ksato
I I'
1701.
= 0.110; p = 0.22
A.
(19)
A more extensive calculation of the line EOF (Fig. 1) gives a curve analogous to EOF in Fig. 2 where A V * = 3.0 kcal. and F* = 1.04 X lo5 dynes/crn. A similar calculation through a pair of lines, such as LM, parallel to EOF and close to it, established the modjt probable reaction path to be only about 0.014 A. removed from the normal mode path a t 500'K., the center of the data. The same line was used to calculate tunnelling a t all temperatures, even though i t is realized that this may slightly underestimate the tunnelling a t low temperatures. The data from the literature for the reactions13of methyl radicals with acetone and ethane to abstract H or D were reviewed extensively in T-I. The vibration frequencies of the activated complex were computed in T-I, and these will be used again. Also in T-I we calculated the tunnelling correction using an Eckart potential along line AOB. In Fig. 12 we see the experimental points, the curve (&4)which includes no tunnelling correction (curve A is dominated by the differences of R-H and R-D in the reactants, and it is very little affected by any assumptions about the activated complex), curve (C) which includes a tunnelling correction based on AOB, as if the mass were constant along this line (r'*r,p.), and curve (B) which is based on eq. 18. The experimental scatter is discouragingly large. However, we feel this analysis may indicate the important factors in large tunnelling corrections in reactions of the type: heavy group-hydrogen-heavy group. The heavy end groups constrain the tunnelling motion to be along a straight line of -45O slope in Fig. 1. A consideration of tunnelling over all the area GLMH in Fig. 1 shows that most of it occu;s over a band such as ELMF, which is about 0.1 A. wide. Tunnelling cannot occur from A to B because of the very high barrier opposing a -45' motion between these points, Tunnelling cannot occur around the curve from A to B because any motion other than the -45' line forces the heavy end groups to move, and their mass is so great that their tunnelling is negligible. By the time the system has moved up to where i t can get a good tunnelling path near EOF, i t has almost gone to the top of the barrier anyhow. Thus we picture the reaction process as activation by collision up to region E or L in Fig. 2, and then a quantum mechanical barrier penetration, reflection or overflight across the area ELMF. Curve (B) in Fig. 12 is based on this model, and i t provides a reasonable representation of the data. Acknowledgments.-We are very grateful to Dr. Ralph Weston for providing us with a complete Sato potential energy surface for Hs. Also we are (13) E. W. R. Steacie, "Atomic and Free Radical Reactions," See. Ed., Reinhold Publishing Corp., New York, N. Y., 1954; J. R. McNesby and A. S. Gordon, THISJOURNAL, 7 6 , 823, 1416 (1954); F. 0. Rice and T . A. Vanderslice, ibid., 80, 291 (1958).
Jan. 5, 1961
LARGE
TUNNELLING CORRECTIONS
indebted to the Alfred P. Sloan Foundation for a Fellowship in support of this study. Appendix I Derivation of Equation 4.-The derivation of eq. 4 follows very closely the derivation of eq. 15a of ref. 6, H JR, except that it is for an activated complex, for .which one separable normal mode is the unstable motion above an inverted parabola between limits -6/2 to 6/2. The classical partition function for this normal mode is
where q is the reaction normal mode coordinate, p is the reaction normal mode momentum, f* is the force constant of the reaction normal mode and is a negative number and m* is the mass associated with the reaction normal mode. Defining the first integral in (21)as A and evaluating the second integral, we get Q* = ( l / h ) A (2am*kT)’h (22) The classical partition function for the entire activated complex (regarded as non-linear and with N atoms) is (cf. H J R , eq. 3 ) N Qolass = Z hs/(2nmak T ) % (23) a = 2
REACTION RATES
9
But the symbols (f*/m*)’/9/2II are just the definition of v*, the imaginary frequency in the reaction coordinate. Upon substitution for v*, we get eq. 4.
Appendix 2 Evaluation of Force Constant Determinant of Activated Complex from a Potential Energy Surface.-From the derivation above, we see that the force constant determinant required in eq. 5B can be obtained directly from a potential energy surface. For a linear three-atom complex and with a R1-R2 plane such as Fig. 1, the stretching force constants about the atom transferred are given directly by various curvatures through the saddlepoint. The coefficient F11 of eq. 1 is given by 2A V = F11ARl1 for a line through 0, parallel to is given by the Rl axis, so that ARz = 0. The coefficient FZZ 2A V = FzzARz2 along a line through 0 parallel to the Re axis. To obtain F ~ zwe , must find the curvature through some line not parallel to R1 or Rz; either COD or EOF is very convenient for this purpose, although other lines through 0 of slope c may be used. If c = dRz/dR1, the general expression for changes in potential energy from the saddlepoint becomes, along this line
+
2 d V = (Fii f C’FZZ 2~Fiz)(dRi)’
(29)
The square of the distance along this line is
where ma refers to atomic masses. The complete partition function in terms of 3 N normal modes (including rotations and translations as normal modes) is ( c f . H J R , eq. 10) 3N Qolasa =
IN
V 8 ~ ~ ( 2 ? r k T / h ~ ) ~IMI’/z’ / 2 j
-7
fi
ui-1
Q*
i = 1
(dy)’ = (dRi)’
+ (dRz)’ = (1 + C’)(dRi)’
(30)
The force constant along this line is
Fc = 2 dE/(dr)’ = (Fl1
+ C’FZZ4-~ c F I z ) /+( ~
6’)
The value of the interaction constant is
where Q* is given by 22. A comparison of eq. 23 and 24 shows that the configurational integral is (cf. H J R , eq. 11) 3N
v8Hzhf3/* 1 J!
Z=
-7
(‘/z
For the special case that F U = FZZ
Ui-’
i = l
Ilmaa/z( 2 a k T)ah’- ~2
u*Q*
(24)
where the product of u is raised from 3 N - 7 to 3 N - 6 by multiplying and dividing by u* = hv*/kT = h(f/m*)l/*/ For the special case of (31) that c = - 1, for example, EOF 2HkT. In a vibrational analysis we again obtain [F.] [Gel = in Fig. 1 3N - 6 F1z = (Fi1 F22)/2 F-1 (33) X i , where F. and G, refer to local valence bond For the special case of (32) that c = -1 and F,, = FZZ i = l properties, and Xi = (2avi)Z. The product of the last FIZ = F,, - F-1 (34) two terms in (24) gives considerable cancellation The force constant determinant in eq. 4 is thus evaluated u*Q* = f * A / ( h k T ) l h (25) From (21) we see that A is a function only off* and T ; it is not a function of mass. Consequently we again have a For a saddlepoint with one negative curvature, the expresseparation of mass and force-constant variable in analogy to sion K I F 2 2 - F,? is negative, and thus its sauare root in ea. 4 (HJ R eq. 13a, b). is imaginary, GancellGg the imaginary faStor in v*. ?he V8a2itla/*11 /‘/2 = ZI F, \‘/z (2akT)l/z factor ws which “diagonalizes” F, is (1 - F I ~ ~ / F ~ (cf. ~Fz~), J, = HJR eq. 16). IIma3/2/G. I’/z (2akT)3N-‘3/2 f*l/zA (“) The G-determinant can be taken from the geometry of the The first identity in J does not depend on force constants, activated complex in the usual way, ref. 7. The-secular the second does not depend on mass, and therefore both the equation [FG - EX] = 0 can be solved for the negative value first and second identities depend on neither. Thus the of A*, without necessity of finding the other roots. From first identity depends on the geometry of the activated com- this value of X*, we find v*, to substitute into eq. 4. The plex as if it were a normal molecule, and the table of J’s in other roots of the secular equation give the frequencies reH J R is directly applicable t o the activated complex struc- quired for eq. 7, if quantum corrections are needed for the ture. The configurational integral for the activated complex real vibrations. IS From this analysis we illustrate that in order t o calculate a rate according to activated complex theory, we do not need the entire potential energy surface but only a small segment of i t near the saddlepoint. If the de Broglie wave length is small compared to the region of separability, that is, or, defining “Z*” as in eq. 5B we have the quadratic region, we use eq. 7 to calculate the rate, and all we need is the location of the saddlepoint and its infiniZ+ = A/(2akT)‘/z tesimal curvatures there. For large degrees of tunnelling, Replacing Q */QA~QBin ( 2 )by Z */ZA=ZBwe get cancellation larger portions of the potential energy surface are needed, as of A , k T , and h discussed in the body of this paper.
+
fl
“z+”f*’/~
-
