S.G. CHRISTOV AND 2.I,. GEOIZGIEV
1748
~
~
~~~
Table I V : EHT Calciilatioris on the Effects of Bending the Hydrogen Bond in Formaldehyde-Water System (+
=
0')
De, 8, deg
0
20 30 60
R," A
eV
koa1 mol-'
-436.95
3.2
ET,
I
2.6
2.6
-436.94
2.7
-436.93
3.2
-436.83
3.0 2.8
0.5
CtlargoSa 00
C*
HC
Sad
cr5.438 (5.519) ? r l ,821 (1.821) cr5.438 ~1,821 ~5.456 ?rl,821 ~5.506 ~l ,821
2.695 (2.690) 0.179 (0.179) 2.694 0.179 2.692 0.179 2.692 0.179
0,380 (0.372)
5.326 (5.257) 2.0 (2.0) 5.326 2.0 5,311 2.0 5.268 2.0
0.0 (0.0) 0.380
0.0 0.379
0.0 0.373
0.0
-----Overlap 0-H
0.3959 (0.4527)
0.1017
(0.0)
0.7609 (0.7573)
0.3963
0,1016
0.7626
0.4089
0,0790
0.7623
0.4408
0.0151
0.7568
Charges on the atoms in the parent electron donors ( D ) and acceptors (A) are shown in the parentheses. carbonyl group. Hydrogen taking part in hydrogen bonding. d X is t,he acceptor oxygen atom. Q
Acknowledgment. The authors are thankful to the Council of Scientific and Industrial Research, India, for
populations-----O.'.H cF=o
* Carbon of
the donor
the support of this research and the staff of the I I T K Computer Centre for the facilities.
On Tunneling Corrections in Chemical Kinetics by S. G. Christov* Institute of Physical Chemistry, Bulgarian Academy of Sciences, Sofia, Bulgaria
and Z. L. Georgiev Department of Physical Chemistry, Higher Chemico-Technological Institute, Sofia, Bulgaria
(Receited August 6 , 1970)
Publication costs borne completely by T h e Journal of Physical Chemistry
A comparison is made between the results of one-dimensional and two-dimensional procedures of calculating tunneling corrections for reactions of the type AH B -+ A HB (via a linear activated complex A-H-E), where A and B are heavy atoms. An improvement of the method of Johnston and Rapp for estimation of
+
+
two-dimensional tunneling corrections for these reactions is proposed. Computations show that the onedimensional treatment of the (extended) reaction path overestimatestunneling, but not too much, but underestimates it, if the mass transferred is taken to be equal to the proton mass.
I. Introduction The role of the tunnel effect in the kinetics of themical reactions of hydrogen and its isotopes has been discussed many times in recent years.'-1° There exist both theoretical reasons and experimental facts which clearly show that it is necessary to introduce a correcting factor t o the classical reaction rate in order to account for the tunneling of protons or hydrogen atoms through the potential energy barrier. An exact solution of the tunneling problem is possible in the usual one-dimensional approximation. Johnston5 has posed the question whether this approximation is justified if one wishes t o connect it with an apT h e Journal of Physical Chemistry, Vol. 7 6 , S o . 11, 1971
plication of the activated complex method for a complete evaluation of the reaction rate. He has stressed (1) R.P. Bell, T r a n s . Faraday Soc., 5 5 , 1 (1959). (2) R . E. Weston, J . Chem. P h y s . , 31, 892 (1959). (3) S. G . Chrlstov, 2. Elektrochem., 62, 567 (1958). (4) S. G. Christov, Dokl. A k a d . S a u k S S S R , 136,663 (1960). (5) H. S. Johnston, A d z a n . Chem. P h y s . , 3, 131 (1960). (6) T. E. Sharp and H. S.Johnston, J . Chem. Piiys., 37, 1541 (1960). (7) H. S.Johnston and D. Rapp, J . Amer. Chem. Soc., 83, 1 (1961). (8) E. I?. Caldin, Discussions Faraday Soc., 39, 2 (1965). (9) S. G. Christov, J . Res. Inst. Catalysis, Hokkaido Unin., 16, 169 (1968). (10) E. M . Mortensen, J . Chem. P h y s . , 48, 4029 (1968); 49, 3526 (1968).
1749
TUNNELING CORRECTIONS IN CHEMICAL KINETICS
that the motion of the system along the reaction path could be treated as a one-dimensional tunneling problem only in the vicinity of the transition state, which practically means a restriction for the application of the usual method to reactions at relatively high temperatures. Johnston and Rapp' have suggested an approximate method which, under certain conditions, allows a consideration of the problem for motion with two degrees of freedom. I n this way they have found that at low temperatures the tunneling correction may be considerably smaller in comparison to that obtained in treating the movement of the system in one dimension along the entire classical reaction path. These results have been already discussed in a previous paper.9 It has been shown, using the numerical data of Sharp and Johnston,Bthat the one-dimensional treatment of the reaction path gives a lower limit of the tunneling correction provided that the mass transferred through the barrier is equal to the proton mass, r n ~ ~ (see Conclusions). Figure 1. Potential energy map of the complex A-H-B (A = = H and B = H); S, saddlepoint; T ~ saddlepoint , Recently MortensenlO has treated quantum-mechancoordinate; OP, bisecting line; RSC, H = ically the isotopic reactions of the type Hz reaction coordinate; KL, MN, and FG, cross sections H Hz as a two-dimensional problem. He has found with a slope of -45". that the tunneling corrections are greater than the onedimensional ones although they do not differ largely from them. that the potential energy of the system T/'(rAH,rHB) is I n view of the importance of the problem, in the presa function of two distances TAH and THB. ent work an investigation on the conditions of apThe topographic map of the potential energy surface plicability of one- and two-dimensional approach for for the system H-H-H has been constructed by evaluation of' the tunneling corrections is carried out, Weston,2 using Sato's method" in a coordinate system and a comparison of their results is made. For this with an angle between the axes of 60". Following purpose, an improved modification of the procedure of Johnston and Rapp7 we shall use Westons's data in a Johnston and Rapp7 is used which is based on a better rectangular coordinate system (Figure 1) at conditions approximation of the potential barrier. A simplified which will be specified below.12 approximate method for calculation of the tunneling The reaction is described classically in the plane corrections is also proposed, which permits one to re?-AH, r H B by the translational motion of a representative place a large part of the numerical calculations by point, with an effective mass 1,13 along the reaction means of analytical expressions. coordinate RC, passing through the saddlepoint S of To verify the conclusions of Johnston and Rapp7 we the potential energy surface V(TAH,THB).I n the vicinuse the same potential energy surface; however, we ity of that point the movement of the system may be refer it to a hypothetical reacting system for which the also treated as an unstable harmonic vibration * procedure of calculation employed is much more justig . . *Bwith an imaginary frequency vs* = l / 2 n d f * / p * , fiable than in the case studied by those author^.^ This where the force constant f* < 0 and the effective mass is permissible as our purpose is to compare one-dimenis given by the expression sional and two-dimensional tunneling corrections under the same conditions.
+
+
A.
11. General Considerations Let us consider the reaction
AH
+ B + A + HB
(1)
where H is a hydrogen atom or proton, and A and B are atoms or radicals treated as mass points. I n particular, A = H (D or T) and B = H (D or T). We assume that energetically the most favorable reaction path corresponds to a linear complex A . . H . . .B, so e
(11) S a Sato, J . Chem. Phys., 23, 59,2465 (1955). (12) It should be stressed once more that for the purpose of comparing the one- and two-dimensional approach for tunneling calculation it is permissible to use a somewhat arbitrary potential energy surface. The conclusions drawn from such comparison will p r o b ably not change essentially if Weston's energy surface is replaced by a more accurate one (see, for example, the work of Shavitt [J.Shavitt, ibid., 48, 2700 (196S)l which shows, however, that the Weston surface is qualitatively correct). (13) The effective mass varies during the movement along the reaction path because of the change in its direction ( c f . eq 5). The Journal of Physical Chemistry, Vol. 76,No. 11, 1971
1750
S. G. CHRISTOVAND
mH, mA, and mB being the masses of H, A, and B, respectively. If A and B are heavy atoms or radicals, so that mH
