HYOUNGMAN KIM
1716
Combined Use of Various Experimental Techniques for the Determination of Nine Diffusion Coefficients in Four-Component Systems* by Hyoungman Kim Institute jor Enzyme Research, University o j Wisconsin, Madison, Wisconsin 69706 (Received November 14, 1968)
The procedures of isothermal diffusion studies of four-component systems2 are extended so that data from different types of diffusion experiments may be combined to evaluate the nine diffusion Coefficients. It is possible to combine data from the same boundary condition as well as from different boundary conditions provided the same intensive property is measured. It appears that certain combinations of the experimental quantities, such as from Gouy and Rayleigh optical measurements of free diffusion, may give improved values of diffusion coefficients compared to the procedures described previously.2 Theoretical interpretations of these general procedures are given.
Introduction The elucidation of the transport mechanisms of various electrolytes through biological membranes, with their important roles in neuromuscular and other physiological functions, is one of the most challenging scientific endeavors of today. I n the steady state with nonequal distributions of potassium and sodium ions across a cell membrane, it is generally believed3 that transport of these ions by intricate biochemical mechanisms, including the hydrolysis of ATP by membrane ATPase,4 is counterbalanced by diffusion along their electrochemical potential gradients through the membrane “pores.’’6,6 Although the study of these biochemical mechanisms is of prime importance, any quantitative study of them is meaningful only after the diffusion process is likewise understood. The diffusion through a biological membrane is an extremely complicated affair. It is, among other things, a multicomponent system where the flow interactions between a large number of components may be significant. Our present knowledge on the flow interactions in multicomponent systems is still very limited even in single-phase systems. h’evertheless, it was shown that the cross-term diffusion coefficients in some of the ternary systems studied are quite ~ignificant.~ It is, of course, erroneous to apply directly the results of multicomponent diffusion studies in an aqueous system with no membrane present to the diffusion through a biological membrane without any qualification; the presence of the membrane component will generally modify profoundly the diffusion coefficients, the modification being made by the charge in the membrane, possible change in water structure, etc. For that matter it is also true that diffusion studies with an artificial membrane, such as an ion-exchange membrane, will not generally explain the diffusional flow within the biological membrane, at least until the detailed structure of the biological membrane is clarified and a strucThe Journal of Phyeical Chemistry
turally similar artificial membrane can be prepared. However, diffusion studies with no membrane present of multicomponent systems involving those ions, other metabolites, and membrane constituents such as lipoprotein in aqueous systems and subsequent transport studies of small solutes through artificial membranes will, undoubtedly, help in understanding the diff usional aspect of the biological transport. It is hoped that procedures presented here will serve that purpose.
General Considerations I n the previous paper,z it was shown that the solutions of the basic differential equations for free diffusion, restricted diffusion, and diaphragm cell measurements on a four-component system have the same general form
where the coefficients qf5are identical for all three boundary conditions and are represented as
(1) This investigation was supported in part by Public Health Service Research Grant AM-05177 from the National Institute of Arthritis and Metabolic Diseases. (2) H. Kim, J. Phys. Chem., 7 0 , 562 (1966). (3) See for example, R. Whittam, “Transport and Diffusion in Red Blood Cells,” The Williams & Wilkins Co., Baltimore, Md., 1964. (4) J. C. Skou, Progr. Biophys. Biophys. Chem., 14, 131 (1964). ATP is adenosine triphosphate; ATPase is the enzyme adenosine triphosphatase.
1717
DETERMINATION OF DIFFUSION COEFFICIENTS IN FOUR-COMPONENT SYSTEMS
GSU,-
ug(n
+ d]} ( j
= 1, 2, 3)
(2~)
in which gl(0) =
Ul2(u3
- u2) + d(ai
- ua) + d
where
-
( ~ z ai)
(3)
Equations 2 are subject to the following restrictions: if ,j = 1, then k = 2 and E = 3; if j = 2, then k = 3 and E = 1; i f j = 3, then k = 1 and E = 2. It is recommended that the reader consult the previous article2 for definitions of the symbols. The 4, are different for different boundary conditions but are all exponential functions of 2, t, and ut (except for the diaphragm cell where the 4, are functions of the ai and t only). As direct determinations of the concentration distributions are not usually possible, some intensive property which is linearly related to the solute concentrations is measured. The relationship between an arbitrary intensive property M and the concentrations of the solutes may be expressed by 3
M = M(G,G,G)
+ C m c ( C i - et) i=l
(4)
where M (GI,C2,C3) is the value of the intensive property of the solution in which the solute concentrations 2‘2, and respectively. Here mi represents are the intensive property derivative, ( h M / h C J C j of the ith solute. Introduction of eq 1 and then eq 2 into eq 4 gives
e’,
el,
M
=
AT
+ -AM2
i-1
rt(?n)@t
(5)
Here A 7 is the intensive property of the solution a t the = C, and* initial boundary position where each
e,
wdm)
=
+
[Xt(m)Mu)
Yt(m)$a(U)- 4 4 ( ~I/gi(u) )
42(.)
=
((73
-
Uz)%
+ + (u1
43(u) = U l ( u 3
- ‘JZ)@l
uZ(‘Jl -
u3)@2
u3)@2
+
(UZ
‘J3(u2
(10)
- ul)4,3
(11)
- Ul)@3
(12)
and $4(4)
=
- u22)% + U2(ui2 - d ) @ z
u1(‘J32
+ Q(d-
CJi2)@3
(13)
In eq 10 both X , ( m ) and Yt(m)are the same for all the boundary conditions if the same intensive property is used and if the mean concentration of each solute is the same in all the experiments. In the procedures described previously2for obtaining the nine diffusion coefficients from measurements using a given boundary condition, three different “experimental quantities” are required for any of the three boundary conditions. Except for the area under the fringe deviation graph in free diffusion, all the experimental quantities are essentially in the form of eq 9. Careful examination of the previous paper2 will show that the derivations of expressions for the experimental quantities from eq 9 involve changes only in the functions 4 in eq 11-13. Since these 4, functions do not contain A c t or Dig, the resulting equations are still (6) There is no direct evidence for the existence of pores in cell membranes although the presence of some sort of gaps in the membrane which are filled with water are suggested indirectly.6 The clarification of the structure of membranes including the pores, whether fixed or transitory, will be of prime importance in understanding membrane transport. (6) A. K. Solomon, J . Uen. Physiol., 43, 1 (1960). (7) The up-to-date references for the ternary systems may be found in ref 2 as well as in, for example, G. Reinfelds and L. J. Gosting, J . Phys. Chem., 68, 2464 (1964). (8) Although there is no true initial boundary position in diaphragm cell diffusion, the general argument presented here is valid.
Volume 73, Number 6
June 1969
1718
HYOUNGMAN KIM
linear in terms of a&) and have intact X,(m) and Y,(m), although in some cases, such as the second moment of the refractive index gradient curve in free diffusion, terms involving either X,(m)or Y,(m) are canceled out. From two of these experimental quantities one may obtain expressions for X,(m)and Y l ( m ) and these are inserted into expressions for the third experimental quantities making these functions of the ai’s. The resulting equations are solved for ut using successive approximations. I n summary, the final quantities derived from the diffusion experiments are the three values each of X,(m), of Y,(m),and of a,,respectively. The solution of eq 7 and 8, the definitions of E,, Ft, G,, and the three relations2 (ai
+ + =DIIDZZ + + DzzDaa - D12Dzi - DiaDai - DzsDaz + + Dii + Dzz + DIlD22D33 f DizDzsDai + Di3DziD3z - DiiDssDaz -
(aiaz
011033
Q~)/V~QQ
~
1
~
a sz ~ ) / a i ~ a= a
(14)
0 3 3
(15)
(Dij/
(16)
l/alaZa3 =
DizDzlDaa
- Di3Dz2D3i
=
give the following expressions for the individual diffusion coefficients
(m1lmJD1t
=
Mm)
I [Ydm) - Yz(m)lGt(m)+ - edm) I +
[E@J Ydm) -
and
I n the procedures described previously, the experimental quantities are obtained only from the same boundary condition using the same experimental technique. This forced one to use some less accurate quantities such as values of various moments for each boundary condition. Since then it was realized that there is no a priori reason why experimental quantities The Journal of Physical Chemistry
from different boundary conditions and from different experimental techniques cannot be combined to calculate the nine diffusion coefficients. As was mentioned earlier, most of the experimental quantities are essentially in the form of eq 9 and any two may be used to solve for Xi(%)and Yi(m)provided that the same intensive property is used and the mean concentration of each solute is kept the same in all the experiments. This will give more flexibility in choosing the experimental quantities.
Some Representative Examples Three procedures will be discussed in this section as representatives of many possible combinations. In the first procedure all the experimental quantities are obtained from one boundary condition. Here data from Gouy and Rayleigh optical systems are combined for the case of free diffusion. The second procedure involves two boundary conditions in which data from free-diffusion experiments using the Gouy diff usiometer are combined with data obtained from diaphragm cell experiments employing an optical system which registers the refractive index of the solution. The third procedure employs all three boundary conditions with electrolyte conductivity measurements being used as the experimental technique. (a) Free Difusion Using the Gouy and Rayleigh Optical Systems. I n the procedure for free diffusion of four-component systems employing the Gouy diff usiometer12the reduced height:area ratios, DA, the reduced second moments, Qm, and the areas under the fringe deviation graphs, Q, are required for computing the nine diffusion coefficients. Of these, the reduced second moments contribute the greatest errors to the calculations because graphical evaluation of the slopes of the fringe deviation graph is i n v ~ l v e d this ; ~ leads to significant uncertainties to the computed diffusion coefficients when the cross-term diffusion coefficients are small. By replacing the reduced second moments with reduced fringe numbers from the Rayleigh optical system, improvement in the diffusion coefficients may be obtained. The Guoy diffusiometers now generally in uselo have built-in Rayleigh optical systems and it is relatively simple to interchange these two optical systems. The Rayleigh optical system in a Gouy diffusiometer is usually used only for obtaining the integral fringe numbers and for this purpose a high quality cylindrical lens is not required. I n the present method, however, the quality of the cylindrical lens may become critical. The most straightforward, and perhaps the most accurate, type of experimental quantity obtained from (9) P. J. Dunlop and L. J. Gosting, J . Amer. Chem. Soc., 77, 5238 (1955). (10) See, for example, L. J. Gosting, E. M. Hanson, G. Kegeles, and M. S. Morris, Rev. Sci. Instr., 20, 209 (1949).
1719
DETERMINATION OF DIFFUSION COEFF~CIENTS IN FOUR-COMPONENT SYSTEMS free diffusion using the Rayleigh optical system is the reduced fringe numberf(j) defined ad1
f(j) =
2j - J
J -
different values of ai@). It is interesting to compare the above equations with eq 9 to show that
2(n - a) An
where J is the total number of fringes and j is the number of a particular Rayleigh interference fringe at position x counted from the homogeneous portion of the upper solution. Symbols n and denote the refractive indices of the solution at the level x and at the initial boundary position, respectively ; An is the initial refractive index difference between the upper and the lower initial solutions. By comparing the above equation with eq 9 one obtains 3
c
f(j) = i = l a@).:(R)
(21)
as
Now eq 24 is rearranged to obtain
where R is the specific refractive index derivative. ) wi(m) in eq 10 and t h e In this equation ~ d f ( Rreplaces ai in eq 11-13 become the probability integrals,
@(dGY)
where y = 4 2 4 1 . For free diffusion $Q(u),&(a), and c#I~(u) may be denoted by r # ~ z f ( ( r ) , 4Bf(a), and +:(a), respectively. The reduced fringe number f(j)is, unlike DA and Q, a function of y, that is, dependent on both time, t, and position, x. From a minimum of three experiments with varying ai@), several values of wi‘(R) for given times are obtained by solving eq 21. When the vertical source slit of the diffusiometer is rotated t o the horizontal position and the cylindrical lens is removed, conversion of the optical system from the Rayleigh to the Gouy system is made simply by replacing the mask in front of the diffusion cell with that for the Gouy optical system. Thus the Gouy and Rayleigh interference patterns may be photographed alternatively during a single experiment. The reduced height:area ratio obtained from the Gouy interference patterns has the relation2
1/dZ = ~ i a i ( R )+ ~ 2 a 2 ( R + ) ~3a3(R)
(23)
+ Yi(R)gS(g) - g4(6)]
(24)
- az)dE + (vi - ddG + ( g 2 - g i ) d &
(25)
where 1
~ c l z
=
----[Xi(R)gz(a)
gda) = (us
gda)
=
(ad
-
az)ull/;
(a1
g4(a)
=
+
- CdU2dG + ( g z - a i ) Q d &
+ +
(d - d ) u l d ; (giz - U 3 2 ) U 2 d G
The values of p i
- rJi2)CsdG
Equation 31 becomes a function of ai only when eq 6, 29, and 30 are introduced; then the successive approximation method described in the previous paper2 may be applied here to evaluate the three values of ui by using three values of Q/da>A obtained from three diffusion experiments a t given mean solute concentrations but with different initial relative solute concen-
(26)
(27) are obtained by solving eq 23 for three (Uzz
d&+da, d&G dGTG
g ( d G d 4 = -___ 242
(11) See, for example, L. J. Gosting in “Advances in Protein Chemistry,” Vol. XI, J. T . Edsall, et al.,Ed., Academic Press Inc., New York, N . Y., 1956; also, H.Svensson and T . E. Thompson, “Translational Diffusion Methods in Protein Chemistry,” in P. Alexander and R. J. Block, Ed., “Analytical Methods of Protein Chemistry,” Pargamon Press, New York, N. Y., 1961.
Volume 79, Number 6 June 1969
HYOUNGMAN KIM
1720
tration differences. Here the computation is somewhat more involved than in the previous method using free diffusion2 because it includes the evduations of the probability integrals @(&y). The evaluation of @ ( d G y ) from given values of ut and y may be conveniently performed by series expansions of the probability integral.12 For small values of
G
Y
=
n=O
uinyn +1
(-lP (2%
+ l)n!
(32)
and for large values of d z y the following asymptotic series may be used
Once the v 2 are evaluated this way for the given three values of w:(R) (for given y values), subsequent successive approximations with other sets of three values of w:(R) (for other y values) will be easy because the ut are already known within the limit of the error of the evaluation, and values of each ut thus obtained from different y values may be averaged. In the course of the successive approximations, a number of X i ( R ) and Y@) values for different y values are also obtained and the average values of these, together with average values of uij may be used for evaluation of Dij using eq 17 after each m is replaced by R. Perhaps the most cumbersome part of this procedure is that the reduced fringe number, f ( j ) ,is y dependent and both t and x must be fixed in at least three diffusion experiments in order to obtain w,(R). I n a Rayleigh diffusion experiment it is common practice that the positions, 2, of individual fringe minima are measured a t given times t since measurement of the fractional part of a fringe at a predetermined position x is difficult. It is also difficult to prefix the times of photographing the Rayleigh fringe patterns because the zerotime corrections vary from one experiment t o another and are not known until after the diffusion experiments. This difficulty may be resolved by the following procedure. I n a given free-diffusion experiment, each fringe minimum (hence each f ( j ) value) has a particular g value. Therefore a number of y values obtained for a given fringe minimum from different Rayleigh photographs taken at different times in the given experiment can be averaged. The averaged y values for different fringe minima in an experiment can be curve fitted using the least-squares method in order to obtain an expression for f(j)as a function of y. Once these relations are obtained from at least three experiments where the mean solute concentrations are kept the same but crt(R) values are different, it is possible to fix y values and The Journal of Physical Chemistry
compute three f(j) values for three experiments. Equation 21 for these three f(j)values may then be solved for three aif(R)’s. There should be no reason why other similar optical systems such as a Jamin interference optical system cannot replace the Rayleigh system, although the latter optical system gives a distinctive advantage over other optical systems because one can obtain both the Rayleigh and the Gouy interference photographs from the same experiment. (b) Combination of the Results from Free-Diffusion and Diaphragm Cell Experiments. This procedure utilizes the reduced height :area ratios and the areas under the deviation graphs from free-diff usion experiments using the Gouy optical system and the refractive index differences between the two compartments in the diaphragm cell. This method is similar to the one described in the previous section and, except for the case when a good-quality cylindrical lens is not available, probably offers no advantage over the procedure employing Gouy and Rayleigh optical systems in free diffusion. No refractometric technique appears to have been employed in diaphragm cell diffusion so far and here we simply use the refractive index differences between two compartments without specifying the optical system. From eq 9 (34)
where An(t) is the refractive index difference between two compartments during the diffusion process and An is the initial refractive index difference. Also, wtd(R)represents eq 10 in which the at of eq 11-13 are @(t/ui)which is defined as @(t/ui) =
(35)
where IC is the characteristic constant of the cell and is obtained by calibration. Equation 34 is formally the same as eq 21 and may be used with eq 29 and 30 to obtain the ui after @(&,y) is repIaced by O(t/ui). The subsequent procedure is the same as in the preceding section. (e> Combination of Results from Free-Digusion, Restricted Diffusion, and Diaphragm Cell Experiments. This procedure combines the specific conductance differences between two fixed levels of the cell in both free and restricted diffusion and between two compartments in the diaphragm cell. Refractometric techniques may be also used but the conductance measurement may be more sensitive, especially in the case of restricted diffusion. In the procedures of Harned, (12) See, for example, J. B. Scarborough, “Numerical Mathematical Analysis,” 3rd ed, The Johns Hopkins Press, Baltimore, Md., 1956, p 391; also “Tables of the Error Function and Its Derivatives,” National Bureau of Standards Applied Mathematics Series, U. S. Government Printing Office, Washington, D. C., 1954, p 41.
DETERMINATION OF DIFFUSION COEFFICIENTS IN FOUR-COMPONENT SYSTEMS
et al.,13 two pairs of electrodes are placed symmetrically 21/3 distances above and below the original boundary position in order to facilitate the conversion of the Fourier series. Here I denotes one-half of the cell height. I n free diffusion, however, the choice of the locations of the two pairs of electrodes is rather arbitrary as long as they are positioned symmetrically about the initial boundary position. From all three boundary conditions one obtains essentially the same expression which derives from eq 9
Here AK(0) is the initial difference in the specific conductance, AK(t) is the time-dependent specific conductance difference, and K denotes the specific conductance derivatives. For restricted diffusion w:(K) represents eq 10 in which @ t are the Fourier series, @(a4,x,t),defined as2
exp[-(2n
+ 1)2a2t/412at] (37)
A minimum of three experiments for each boundary condition at the same mean solute concentrations will give values of a total of nine w ( ( K ) , which may be represented as Wf(K) =
W:(K)
=
+
[xi(K)4zf(C) Y h ) 4 a f ( d - 44f(a)I/gda)
(38)
-k Yi(K)43r(r)- 44r(C) l/gl(al
(39)
[xi(K)$;(C)
and d
ut ( K )
=
+
[xt(K)4zd(u) Yi(K)r7hd(a)44d(a)I/gi(a, (i = 1, 2, 3) (40)
where 4 ( a ) are defined in eq 11, 12, and 13. The difference in the 4 ( a ) for the different boundary conditions occurs only in the functions @. It should be remembered that the function @(d&) for the present purpose is only dependent on time. When two pairs of electrodes are located at x = *221/3, as is the present case, for large values of t, eq 37 can be simplified to get
(41) where r = ?r2t/412
(42)
and &‘(u), +:(a), and 44r(u) become identical with f ( a , r ) , g(a,r), and h(u,r), respectively. These latter functions are defined in eq 68 of ref 2. For diaphragm cell method 4 z d ( u ) , +?(a), and $dd(u) become identical with f(o,kt), g(a,kt), and h(a,kt) in eq 72 and 73 of ref 2.
1721
After obtaining values for the u i f ( ~ u) ,t r ( ~ )and , eq 36 by using experimental data, it is possible to solve any two of the three equations (eq 38, 39, and 40) for X , ( K )and Y , ( K )and introduce the resulting equations into the third equation to make this equation a function only of the ut. The successive approximation method described previously will give ) from these the values for ut, X , ( K ) ,and Y I ( ~and nine diffusion coefficients may be computed using eq 17 provided that K values are independently determined. I n order to obtain three values of w ~ ( K ) in each boundary condition at least three diffusion experiments with varying ai values have to be performed and the specific conductance data at the same t value have to be used for solving eq 36. As was mentioned previously, fixing the time for the measurements is not an easy task because the zero-time corrections vary with experiments. If continuous records of the specific conductance difference, AK(t),are made, this difficulty may be overcome. It is not required t o fix the time when the values of W ( K ) from three bounGary conditions are combined to compute the nine diffusion coefficients although this will be convenient for the successive approximations. Unless the restricted-diffusion cell is very short, it is not.practica1 to measure the AK(t) values in free and restricted diffusions during the same time interval. u i d ( ~from )
Discussion There are a number of other possible combinations with lesser practical applicability. As an example, combination of the reduced second moment from free diffusion employing the Gouy optical system and the zeroth and first moments of refractive index differencetime curve from diaphragm cell diffusion (eq 36, 48, 78, and 79 of ref 2) makes the computation somewhat simpler. An attempt to combine the specific conductance measurements and the Gouy optical system in free diffusion was unsuccessful. The essence of the procedures described here as well as in the previous paper is that a minimum of nine “experimental quantities” are solved for three X,(m), three Y,(m), and three ai, and these, in turn, are solved for the nine diff usion coefficients. This requires that each X,(m), Y , ( m ) , and ai from different experiments be identical. This requirement is met even when the experimental quantities are obtained from different boundary conditions using different experimental techniques as long as the same intensive property is used. If different intensive properties are used, however, X,(m) and Y,(m) from different experiments are no longer identical making the proposed procedures inapplicable. This general method of combining different experimental techniques should also apply to the isothermal diffusion studies of ternary systems. Because this (13) H . S. Harned and D. M. French, Ann. N . Y . Acad. Sci., 46, 267 (1946); H. S. Harned and R. L. Nuttall, J. Amer. Chem. Soc., 69, 736 (1947).
Volume 79, Number 6 June 1969
1722
DONALD G. TRUHLAR AND ARONKUPPERMANN
method will not generally improve the existing procedures, further discussion along this line will not be given. The increased flexibility in the choice of the experimental quantities opens the way to the determinations of 16 diffusion coefficients from isothermal diffusion studies of the five-component systems. There are, however, several factors one must resolve before undertaking this task : accuracy of the experimental
quantities available, the computation time required for the evaluation of four at's, and, in view of the tedious nature of the experiments involved, a strong motivation. Perhaps some experiences from diffusion studies of the four-component systems will shed light on these factors. 'Acknowledgment. It is a pleasure to thank Professor L. J. Gosting for helpful discussions and for his criticism of the manuscript.
Application of the Statistical Phase Space Theory to Reactions of Atomic Hydrogen with Deuterium Halides18 by Donald G.Truhlar and Aron Kuppermann Arthur A m o s Noyes Laboratory of Chemical Physics,lb California Institute of Technology, Pasadena, California 91 100 (Received November 20, 1968)
A statistical phase space theory of chemical kinetics, developed by Light, Pechukas, and Nikitin, is used to predict the reaction cross sections and rates of the processes H AB -+ HA B and HB A, where AB = DCl, DBr, DI, and HI. The theory contains no adjustable parameters and agreement with experiment for the abstraction fraction [HA]/( [HA] [HB]) is fair. The predicted rates are too large, indicating that the statistical assumption is not completely valid. The predicted dependence of the reaction probabilities on initial orbital angular momentum and the predicted internal energy distributions of the products are also presented and discussed. For AB = DI, the prediction for the fraction of I atoms produced in the 2Pi/,electronically excited state is in disagreement with recent experiments.
+
+
+
+
I. Introduction A statistical phase space theory of binary rearrangements requiring little knowledge of the interaction potentials of the reaction system except at large separations of stable reactants and products has recently been developed by Light and Pechukasa2 Since then this theory has been applied to several ion-neutra13-' and neutral-neutral collision processes.s-10 The applications of the theory have been reviewed by Light. l1 Some other calculations and theories of chemical collisions using phase space and statistical approaches which do not assume detailed knowledge of the potential energy surface near a tight transition state have been presented by Keck, Hoare, Firsov, and Marcus.I2 The statistical theory is usually expected to be good for predicting product ratios, product rotational energy distributions, and the energy dependence of reaction cross sections.6-l1 The reactions H DBr, H DI, H I present an opportunity to compare the and H statistical theory with experiment.lG15 The reaction of H with DX can lead to either H D X (abstraction) or H X D (exchange). Photochemical experiments
+
+
T h e Journal of Physical Chemistry
+
+
+
in these l a b ~ r a t o r i e s ' ~have ~ ' ~ given values for the abstraction fraction of products
(1) (a) This work was supported by the U. S. Atomic Energy Commission, Report Code No. CALT-532-29; (b) Contribution No. 3670. (2) (a) J. C. Light, J . Chem. Phys., 40, 3221 (1964); (b) P. Pechukas and J. C. Light, ibid., 42, 3281 (1965). (3) E. E. Nikitin, Teor. i Eksperim. K h i m . Akad. N a u k Ukr. SSR, 1, 135 (1965); Theor. Ezptl. Chem., 1 , 83 (1966). (4) E. E. Nikitin, Teor. i Eksperim. K h i m . Akad. N a u k Ukr. S S R , 1 , 428 (1965); Theor. Erpt2. Chem., 1 , 276 (1965). (5) J. C. Light and J. Lin, J. Chem. Phys., 43, 3209 (1965). (6) L. M. Tannewald, Proc. Phys. SOC.(London), 87, 109 (1966). See T. F. Moran and L. Friedman, J . Chem. Phys., 45, 3837 (1966), and R. W. Rozett and W. 5. Koski, ibid., 48, 633 (1968). (7) F. A. Wolf, ibid., 44, 1619 (1966). (8) P. Pechukas, J. C. Light, and C. Rankin, ibid., 44, 794 (1966). We use the notation of this paper wherever possible. (9) J. Lin and J. C. Light, ibid., 45, 2545 (1966). The authors are grateful for a preprint of this article. (10) K. Yang, J. D. Paden, and C. L. Hassell, ibid., 47, 3824 (1967). (11) J. C. Light, Discussions Faraday Soc., 44, 14 (1968).
