1424
NOTES
On the Calculation of the Vibrational
wz(E*)t, where wl(E*) = -f”(E*)/f”’(E*) wz(E*) = (6ifi/j”’(E*))’I*,we obtain
De-excitation Probabilities
by H. Shin*
and
3fi [f”’(E*)1’
Department of Chemistry, Cornel2 University, Ithaca, New York (Received August 26, 1964)
exp{i [ifit3- p(t*)t]} dt (2) C’
where In the evaluation of the essential part of the vibrational de-excitation probability per unit time, P , it is a common to use the Laplace method’ assuming there is a most probable value E * of the initial relative translational energy E for the transition. Use of this method hm been concerned only with the secondorder term in the expansion of the exponent of the relavant integral around E = E*, however, and the contribution of the terms higher than the second order have not been subjected to a critical investigation. In this note we discuss the correction terms which result from these higher-order terms and how significant they are compared to the leading term. For this study we assume the repulsive interaction V ( s ) = A exp ( -s / a ) . The de-excitation probability per unit time is obtained from the transition probability per collision p(E) by P
=
la
Lrn
so that we can write P
= a:
exp(f(E)/fi) dE, where
a: is the composite of the pre-exponential part in p(E) and Z(E). For small 6 , P can be given in terms of an integral representation
P
= a:
s
exp[ if(E)] d E
where c is some contour in the E plane. To determine P for l/fi >> 1, we adjust the contour so that Re f(E) is everywhere as small as possible. Then the largest value of Re f ( E ) throughout the contour will be a t a saddle point for f(E). Near this point f(E) = f(E*) ‘/,f”(E*)(E - E*)’ ’/sf”’(E*)(E - E*)3 . , where E* is the solution of df(E)/dE = 0. Changwl(E*) ing the variable of the integral to E = E*
+
The JournaE of Physical Chemistry
[
6ili
J”’
f”’(E*)
- f(E*)
fi
+ & (f”(E*))3/(f”’(E*))2 ‘ l
We now evaluate P by substituting an explicit expression for p(E), ie., f(E) and its derivatives in the above expressions, which is obtained in terms of the WKB wave functions.4,* For V ( s ) = A exp(-z/a), this approach gives9
*
Department of Chemistry, University of Nevada, Reno, Nev.
(1) L. Landau and E. Teller, Physik. 2. Sowjetunion, 10, 34 (1936). (2) R . N . Schwarts, Z . I. Slawsky, and K . F. Hersfeld, J. Chem. Phys., 20, 1591 (1952); R. N. Schwartz and K. F. Herzfeld,ibid.,22, 767 (1954).
C
,
[f”(E*)I2 2 [f”’(E*) ]
Therefore, the exponent of P is I
+
=
Here e’ is a new contour in the t plane, and t* is the saddle point. In this integral Re (2fit3 - p ( t * ) t ) is monotonic along the path, except at the saddle point, and we may apply the Laplace method to evaluate it asymptotically a t t* = -(p/3ifi)”’. Use of this procedure then simply gives
p(E)Z(E) dE, where Z(E)dE is the number
of collisions per unit time suffered by each molecule, in which the initial relative translational energy is dE. Both p ( E ) and Z(E) are between E and E predominantly controlled by their exponential parts
+
dt*)
+
+ +
(3) E. E. Nikitin, Opt. i S p e k t r o s k o p i y a , 6 , 141 (1959). (4) €3. Widom, Discusawns Faraday SOC.,3 3 , 37 (1962). (5) J. T. Vanderslice and S. Weissman. J . Chem. Phys., 37, 2247 (1962).
(6) D. Rapp and T . E. Sharp, ibid.,38, 2641 (1963). (7) N. G. de Bruijn. “Asymptotic Methods in Analysis,” 2nd Ed., North-Holland Publishing Co., Amsterdam, 1961, Chapters 4 and 5. (8) L. Landau and E. M. Lifshitz, “Quantum Mechanics.” Pergamon Press, Ltd., London, 1958, pp. 178-183. (9) H . Shin, Can. J . Chem., 42, 2351 (1964); J . Chem. Phys., 42, 59 (1965).
1425
NOTES
Carbon-13 Chemical Shift Viewed as a Constitutive Property. 11.
Substituted
Hydrocarbons
where p is the reduced mass of the colliding molecules and A (>O) is the magnitude of the change in the oscillator’s energy owing to the transition. From eq. 4 we obtain E* = (~/2)”3(aaAkT/fi)z’/” - A/2 and
[I -
;);(
+
g
($)2
- ..
7 (”) + 24 E (A)2 E* -
[I - 4 E*
.]
...I
by George B. Savitsky, Robert M. Pearson, and Keishi Namikawa Department of Chemistry, D’nieersity of California, Davis, California (Receized September 16- 1964)
It has been shown’ that CI3 chemical shifts in simple unsubstituted hydrocarbons can be calculated by constitutive additivity with 2 p.p.m. standard error. Although substitution of hydrocarbons by polar groups leads to significant deviations from the constitutive rule, it is possible that these deviations may become useful and theoretically important parameters, as deviations from other constitutive properties, e.g., exaltations of molar refraction. The purpose of this paper is to present some patterns and trends in these deviations which begin to emerge from the available experimental data.
Experimental When these expressions are substituted into eq. 2 and 3, with the aid of eq. 4, we obtain t,he final expression for P,in the asymptotic limit fi -+ 0
(5)
If this expression is compared to the previous obtained by neglecting the higher-order terms in f ( E ) , it can be readily seen that - (4/25)(p/2)’/’((aaAkT/ h)’/’/kT is the correction term resulting from the inclusion of these neglected terms. Therefore, the leading term of the exponent in these previous works is about 5.375 overestimated. Since for various molecules the leading term is 1530 a t ordinary temperatures,2s10taking account of the higher-order terms in f ( E ) may reduce the de-excitation probability by a factor of 2 to 5. Therefore, although the inclusion of such terms does not seriously change P, it is still an important factor to be concerned in any critical comparison between theories and experiments.
The C13 chemical shifts were measured according to the procedure previously described. All chemical shifts are reported in p.p.m. with respect to benzene (64.9 p.p.m. are subtracted from literature values originally reported in p.p.m. from CS2). All commercial reagents were checked for purity. Determination of Polar Bond Contributions to C13 Chemical Shifts. Assuming that the contribution of C-H bonds to C13 chemical shifts is not affected by polar substituents, the constitutive contribution of a polar bond -X to the CI3 shift can be determined by subtracting 3 times the value of the -H bond parameter’ ( i e . , 95.4 p.p.m.) from the C13 chemical shift of the corresponding CH3X. Some values of -X bond paranieters thus obtained are listed in Table I. The =O bond parameter of the carbonyl oxygen was obtained by subtracting the values of -CH3 and -H bond paraTeters’ from the chemical shift of acetaldehyde. Monosubstituted Alkanes. It was previously shown2 that in the monosubstituted alkane series CH3X, C2H5X, i-C3H7X,and t-C4H9X,the effect of replacement of the hydrogen atoms bonded to the a-carbon by methyl groups on the shift of that carbon varied significantly with the nature of X. This, in fact, implies (1) G. B. Savitsky and K. Namikawa. J . P h y s . Chem.. 6 8 , 1956
(10) A. V.‘ Kleinberg and A . N. Terenin, Dokl. A k a d . N a u k SSSR, 101, 1031 (1955).
(Ig6*). (2) G. B. Savitsky and K . Namikawa, ibid., 67, 2430 (1963).
Volume 69,Number 4
A p r i l 1966
