532
Vsl. 66
HAROLD 8.JORX\~~TON AND JULIAN HEICKLIW
would be constant, and would mean only that D as reported here was not a true monomer distri-
bution coefficient, Acknowledgment,-We
are grateful to the Re-
search Corporation for a Cottrell Grant, and to E. I. du Pont de Nemours & Company for departmen tal grants which were used partially for support of this work.
TUNNELLING CORRECTIONS FOR UNSYMMETRICAL ECKART
POTENTIAL ENERGY BARRIERS BY HAROLD S. JOHNSTON AND JULIAN HEICKLEN Department of Chemistry, University of California, Berkeley 6,California Reoeivsd October $0, 1981
Tunnelling corrections have been evaluated for the unsymmetrical Eckart potential for ranges of parameters expected for ordinary chemical reactions at ordinary temperatures.
I n computing chemical reaction rates using activated complex theory, one must include a correction for quantum mechanical barrier penetration and non-classical reflection, the effects referred to as "tunnelling." For small degrees of tunnelling, a correction was derived by Wigner.' Bell2 worked out the tunnelling problem for a truncated parabola and a Boltzmann distribution of incident systems. Shavitta and Johnston and Rapp4 computed tunnelling corrections for symmetrical Eckarts functions for chemically interesting values of the parameters involved. The present article presents similar calculations for the unsymmetrical Eckart function. Eckart's one-dimensional potential energy function is y =
- exp(2r x/L)
(2)
where 5 is the variable dimension and L is a characteristic length. For the symmetrical function, A is zero. Both a symmetrical and an unsym-
metrical Eckart function are given by Fig. 1. It is seen to be flat at both - w and c o s The maximum value is AVi above the value a t m and AV'z above the value a t m. F" is the second derivative of the potential energy function evaluated a t its maximum. The parameters A , B, and L in eq. 1 and 2 are related to AV1, AV,, and F" of Fig. 1by
+-
+
A
AVI
-
(3)
AVg
B = [(AV,)'/l
+ (AV1)'/:I9
(4)
The inverse relations are
+ B)'/4B
(A
AVi AVz
(A - B)a/4B -F* = rZ(A2 Bz)/2LSBa
-
(6) (7) (8)
A particle of mass,m and energy E approaching the barrier is characterized by the parameters u", aland a2 u* = hv'/kT
P-
Y*
=
a1
E
a2
E
(9)
(1/2r)(-F*/m)'/g
(10)
2nAVi/hv* 2rAVn/h~*
(11) (12)
In these variables, the probability K ( E )that a particle starting toward the barrier with energy E a t - a will pass the barrier and appear later at m with energy E is found by solving Schroedinger's equation for the Eckart function, and thc transmission probability is6
+
\
\ \
\
3
'.---.
cash 2 r ( a
K(E) = 1
where
- b ) f cash 2nd
- cos11 2r(a + b ) f
-+-
27fa = 2[a,E]'/2 (a1-1/2
-2
-1.0
-0.5
0.5
0
1.0
A.
Fig. 1.-Symmetrical and unsymmetrical Eckart function. E is one example of the variable energy considered in eq. 13. V* is the same as AV1. F* is dV/dzZ evaluated a t the maximum. (1) E. Wigner. 2. physik. Chem. (Leipzig), Bl9, 203 (1932). (2) R. P. Bell, Trans. Faraday SOC.,66, 1 (1959). (3) I. Shavitt, Theoretical Chemistry Laboratory, University of Wisconsin, Madison, Wisconsin, Report AEC-23, Series 2, 3 (1959). (4) H. S. Johnston and D. Rapp, J . Am. Chsrn. SOL,83, 1 (1961). ( 5 ) C. Eckart. Phys. Rev., 36, 1303 (1930).
cash 2nd
(14)
a2-1/2)-1
2 r b = 2[(1 f $)a1 - a4'/z(a1-'/1 2 ~ = d 2[alaZ- 4rz/16I1/a 4 = E/AVi
+
(13)
a2-'/2)-1
(15) (16)
(17)
When d is imaginary, the function cosh 2nd in eq. 13 becomes cos 2 4 dl. The tunnelling correction factor r*is interpreted as mechanical rate r * -- quantum classical mechanical rate
With a Boltzmann distribution of incident particles
March, I962
TUNNELLING CORRECTIONS FOR UNSYMMETRICAL ECKART POTENTIAL
533
TABU I COMPUTEID BARRIHB PENETRATION QUANTUM CORRECTIONS r * FROM U N ~ Y M ~ T RECHART I C A ~ BARRIERS ~ AS A FUNCTION 08 #I, In, AND Id* U
ai
0.5
1
2
4
8
12
16
20
L
as
2
3
4
5
6
8
10
0.5 1 2 4 8 12 16 20
1.16
1.25 1..21 I.. 14 1..07 1. 00 0.97 0.95 0.94 1.43 1.35 1 ;24 1.I6 1.12 1-10 1.08 1.58 1.47 1.36 1.32 1.29 1.27 1.58 1.51 1.47 1.44 1.42 1.56 1.54 1.53 1.51 1.5 1.5 1.5 1.5 1.5 1.5
1.34 1.29 1.20 1.11 1.03 0.99 0.97 0.95 1.62 1.51 1.37 1.26 1.21 1.18 1..16 1.91 1.77 1.61 1.54 1.50 1.47 2.02 1.93 1.86 1.81 1.78 2.04 2.04 2.02 2.00 2.1 2.1 2.1 2.1 2.1 2.1
1.44 1.38
1.55 1.47 1.34 1.22 L,11 1.06 1.02 1.oo 2.09 1.93 1.71 1.54 1.46 1.42 1.39 2.90 2.66 2.36 2.23 2.15 2.10 3.69 3.56 3.39 3.28 3.20 4.54 4.68 4.65 4.61 5.2 5.4 5.4 5.7 5.9 6.1
1.80 1.68 1.51 1.35 1.21 1.15 1.11 1.OB 2.72 2.50 2.16 1.92 1.81 1.75 1.70 4.55 4.20 3.65 3.41 3.27 3.18 7.60 7.57 7.16 6.88 6.68 13.8 15.4 15.6 15.5 22 25 26 32 37 46
2.09 1.93 1.71. 1.50 1.34 1.26 1.22 1.19 3.56 3.26 2.78 2.43 2.28 2.20 2.14 7.34 6.85 5.87 5.44 5.20 5.03 17.3 18.0 17.0 16.2 15.7 57.0 71.7 74.4 74.2 162 220 246 437 616 1150
1 2 4
8 12 16 20 2 4 8 12 10 20 4 8 12 16 20 8 12 16 20 12 16 20 16 20 20
1.13 1.09 1.04 0.99 0.96 0.94 0.93 1.27 1.21 1.14 1.OS 1.06 1.04 1.03 1.32 1.26 1.19 1.16 1.14 1.12 1.30 1.25 1.22 1.20 1.19 1.24 1.22 1.21 1.20 1.2
1.2 1.2 1.2 1.2 1.2
.
1.27 1.16 1.06 1.02 0.99 0.87 1.83 1.71 1.53 1.39 1.33 1.29 1.26 2.34 2.16 1.93 1.84 1.78 1.74 2.69 2.56 2.46 2.39 2.34 2.94 2.96 2.93 2.90 3.1 3.1 3.1 3.2 3.2 3.2
12
16
2.42 2.22 1.94 1.69 1.49 1.40 1.35
1.31 4.68 4.28 3.60 3.12 2.91 2.80 2.72 12.1 11.4 9.69 8.94 8.51 8.22 42.4 46.7 44.0 41.9 40.3 307 445 473 474 1970 3300 3920
... ... ...
a t - m in Fig. 1, the correction factor can be expressed as
3.26 2.94 2.53 2.16 1.88 1.76 1.68 1.64 8.19 7.48 6.16 5.25 4.88 4.66 4.52 34.0 33.4 28.0 25.6 24.2 23.3 304 376 354 335 321
..
.. .. ..
.. ..
..
..
..
..
that if a1 and a2 are interchanged, the value of r*is unaffected. In ref. 4 and 6, it was emphasized that for chemical reactions, AVI is not the activation energy. The one-dimensional reaction coordinate does not With the transmission function given by eq. 13, extend from the activated complex to the products we have numerically integrated eq. 18 by means of or reactants. The reaction normal mode, when the IBM 704 computer a t the Berkeley Computing extended beyond the quadratic region near the Center. The results can be expressed in terms of saddlepoint, soon encounters "side-wall-repulsions" al, a2and u",eq. 9-12. For chemically interesting on the two-dimensional potential energy surface. values of these parameters, the quantum correction I n fitting an Eckart potential to a potential energy factom aire listed in Table I. It can be seen that surface for a chemical reaction, - OD is not at refor a given a1 and u",the tunnelling correction is actants and m is not at products; rather in terms moderately sensitive to the value of az. One would of the potential energy surface, - m is a cirque make a fairly large error, in most cases, if one used immediately adjacent to the col and 03 is a similar the symmetrical Eckart correctionBS4for a n un- cirque on the other side of the col. symmetrical case. It should be further noted5 (6) Ii. S. Johnston, Advances in Chem. Phys., 3, 131 (1961).
+
+
