3613
CALCULATIOIVS OF SECONDARY CP-DEUTERIUM ISOTOPE EFFECTS
Model Calculations of Secondary a-Deuterium Isotope Effects in kdditioii Reactions to Olefinic Double Bonds
,
Safarik and 0.P. Strausz"
Depertmenl of Chemistry, University of Alberta, Edmonton, Alberta, Canada
(Received April 17, 19793)
Publication costs assisted by the h'ational Research Council of Canada
I n detailed calculations within the framework of transition state theory it is shown that in the addition of methyl or trifluoromethyl radicals t o ethylene, as in the previously reported S(aP) 3. ethylene system, the maw single contributing factor to secondary a-H/D isotope effects is, contrary to earlier notions, not the rehybridization of the carbon atoms but the creation of new normal modes during the reaction. Although calculations thus far have been carried out only for three systems, the effect is clearly independent of the nature of the reagents and in general i t may be stated that a t least one of the newly created vibrations will be isotopically sensitive and generate a secondary a-HID kinetic isotope effect larger in magnitude than that generated by the rehybridization of the carbon atoms.
The secondary a-deuterium isotope effect in addition reactions involvhg olefins has been a t t r i b ~ t e d l - ~ to changes in vibrational frequencies accompanying the partial rehybridization of the reactant carbon atom during passage from the reactant to the transition state. The earlier attempts by Seltzer1 and by Szwarc, et G Z . , ~ to reproduce theoretically the experimental values of thns isotope effect were based on the intuitive extension arid generalizaxion of Streitwieser's original postulate6 in which the main cause of secondary aiaotope effect is the frequency increase in one of the out of plane 6 - H vibrations on the central carbon atom during transition from the sp2to the sp3configuration. 'The experimentally obtained values of kD/kH for the secondary a-dueterium isotope effect in addition reactions to o1t:finic double bonds lie between 1.07-1.14 indicating an inverse isotope effect. In an earIier theoretical study from this laboratory' on the secondary a-deuterium isotope effect in the addition resclrion of S(T) atoms with ethylene an important factor contributing t o isotope effect was uncovered, the significance of which has been completely overlooked before. It was found that, contrary to currently accepted notions, the main single source of isotope effect in this reaction is not the relatively small change in the force constants of the reacf ant accornpariying t,he rehybridization of the central carbon atom lout the creation of new, isotopically sensitive vibrations in the activated complex by the formation of the new bond, corresponding to a relatively large change in the force field by introducing additional force con stank. Furthermore, rclying on a model for the transition state that was derived. on the basis of molecular orbital theory it was possible from a comparison of the computed and measured values of the isotope effect to
conclude that the transition state lies about half way between reactant and product in terms of geometrical changes. On the other hand Szwarc, et ~ l . on , ~ the generally accepted premises that the cause of the secondary aisotope effect is the rehybridization of the olefinic carbon atom concluded from the small value of the measured isotope effect in the addition of CHa and CF3 radicals to olefinic double bonds that the bond between the radical and the carbon atom is relatively long, consequently the extent of the rehybridization is small and the reaction center retains its planar structure in the transition state. Similar considerations led Van Siclrle, et G Z . , ~ t o analogous conclusions regarding the Diels-Alder reaction between cyclopentadiene and maleic anhydride. I n order t o assess the generality of the isotope effect caused by the creation of new vibrations in addition reactions and to examine the structure of the activated complex q7e have performed model calculations for the reactions CHzCHz X -+ * CHzCRzXand CDZCD~ X -+ +CD2CDZX,where X is a group or atom with massof 15(CH3),32(S), and69(CF3).
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+
Results The isotopic rate ratio k ~ / was k ~ calculated by the (1) S.Seltzer, J.Amer. Chem. Soc., 83, 1861 (1961). (2) M.Talrahasi and R. J. Cvetanovi6, Can. J . Chem., 40, 1037 (1962). (3) M. Feld, A. P. Stefani, and M. Szwarc, J. A m e r . Chem, Soc., 84, 4451 (1962). (4) D. E. Van Sickle and J. 0. Rodin, ibid., 86,3091 (1964). (5) W. A . Pryor, R. W. Henderson, R. A. Patsiga, and P;. Carroll, ibid., 88, 1199 (1966). (6) A. Streitwieser, Jr., R.H . Jagow, R. C . Fahey, and S. J. Suzuki, ibid., 80, 2326 (1958). (7) 0. P. Strauss, I. Safarik, W.B. O'Callaghan, and H. E. Gunning, ibid., 94, 1828 (1972).
The Journal of Physical Chemistry, Vol. '76,N o . 84, 1978
I. SAFARIM AND 0. P. STRAUSZ
361 equgtion derived from transition state theorystg
A2
I
Model A
(1)
where the dagger signifies the transition state, and the numbers 1 and 2 refcr to the light and heavy molecules, respectively. 1 ' s are the principal moments of inertia, M's the molecular masses, pZ = hcw,/lcT and w z is a normal vibrational frequency. Equation 1 has usually been abbreviated as a product of three factors ICps/i%o =
Figure 1. Illustration of geometries for the two models of the activated complex.
Table I: Values of the MMI Factor Calculated for the Two Models of the Activated Complex
(KW)(EXC) (ZPE) MX
where MMI, EXC, and ZPE represent the contributions from thc tradational. and rotational energies, vibrational energies, and the zero point energies, respectively. The configuratims of the models used for the activated complex are shown in Figure 1. I n model A, the planar ethylene structure is completely preserved; the 6-C-X angle is 100" and the C-C-X plane is perpendicular to the plstne of the ethylene. This model represents a situation where formation of the C-X bond in the activated complex causes only negligible deviation from the ethylene structure and approximates that suggested by Sz,vvarc, Van Sickle, and coworkers. I n model B the progress of the formation of the C-X bond causes a partial rehybridization of the C atom, accompanied by structural changes in the activated complex; the distance of the C-C double bond is increased compared t o that of model A, and the two C-I3 bonds on the C titom are bent by 10" from the C2H4 plane. I n this model we assumed that the formation of the new bond is accompanied by a gradual progression of the geometry from initial to final state. Values of the three factors in equation 1 were computed individually for the two models and three masses is,32, and 69 of the attacking radicals. The data for the MI'II factor are given in Table I. It is seen that the values are larger than unity and they are affected but little by the model chosen or the mass of the attacking radical. For the calculation of the values of the EXC and ZPE factom the normal vibrational frequencies of the activated comolex are needed, therefore we have perThe Journal OS Physical Chemistry, Val. 76, N o . 24, 1972
Model A Model B a
9-X
=
2.0 A.
Mx
= 15
1.4W
1.434a 1 . 423b
C-X
=
1.7 A.
=- 32
1.453 c
C-X = 2.1 A.
M x = 69
I . 426" 1.417% d
C-X
=
1.8 A.
formed normal coordinate analyses on our models. The computer program used for the iiumerical calculations has been described el~ewhere.7~~ The 6-X stretching mode was taken as the reaction coordinate by setting the C-X stretching force constant equal to zero, as is usual in model calculations when the po.~ tential surface for the reaction is not k n o ~ r ~Assumption of some negative value for this force constant would have no bearing on the result since the imaginary frequencies would be insensitive to deuterium substitution. I n model A, we used the force constants reported in the literature for the ethylene molecule,1° and for the H-C-X and C-C-X bending force constant 0.1 and 0.2 mdyn A, respectively. I n stable molecules, the bending force tonstants for H-6-42 and H-C-S are about 0.6 mdyn A, and those for 6-C-C and 6-C-S about 0.9-1.1 mdyn A; therefore, the H-C-X and C-6-X force constants in model A were assumed to be small in order to indicate a very weak bond be(8) J. Bigeleisen and M. Wolfsberg, Advan. Chem. Phys., I, 15 (1958). (9) M . Wolfsberg and M. 3. Stern, Pure A p p l . Chem., 8 , 225, 325 (1964). (10) B. N. Cyvin and S. J. Cyvin, Acta Chem. Scand., 17, 1831 (1963).
CALCTLATIOKS OF SECONDARY CX-DEUTERIUM ISOTOPE EFFECTS tween C and X and consequently a negligible perturbation of the configuration of the C atom. I n model B some of the force constants were altered in accordance with the structural changes. For the longer 6-C bond t,he force constant was lowered from 10.8 to 8.0 mdynlh, and because of the stronger C-X bond, dhe force constants for the C-C-X and H-C-X bending motion,s were increased from 0.2 to 0.4 mdyn 8 and from 0.3 to 0.2 mdyn b,respectively. Recent quantum chemical calculaxions" have indicated a decrease in the rotational barrier for the terminal methylene group in the activated complex and we accordingly lowered the torsional force constant in model B from 0.54 to 0.34 mdyn 8. The mass of X was taken to be 15 in both models, thus representing a structureless methyl group. Variation of the mass of X from 15 to 32 and 69 resulted in negligible small changes of the frequencies used in the calculation cf the isotope effect. The normal vibrational frequencies obtained from these calculations, with the corresponding values of the EXC and ZPE factors a t 400"K, are collected in Table I1 for model A and in Table IIX for model B, respectively. For comparison, related frequencies of the ethylene molecule are also listed.
Table I T : Values of the EXC arid ZPE Factors Calculated for Model A of the Activated Complex ---VI H
bration, em -I----
\
H
/"a
H
c:-c
X H
\ /-,I/
,c=c, H
H
H
EXC
ZPE
(EXC) X (ZPE)
3615
Table TI1 : Values of the EXC and ZPE Factors Calculated for Model B of the Activated Complex
___-
Vibration, cm-l---Ha / E
H \
c-c
I*
1623
1644
0.999
1.010
1.010
CHz deform
1342 1443
1375 1448
1.001 1.000
0.981
0.982 0.995
CHz rock
1236 810
1240
1.000
1.002
810
1.000
1.000
1.002 1.000
CHI wag
943 949
980 1036
1.001 1.005
0.969 0.949
0.970 0.954
CH? twist
1027
944 380
0 995 0.874
1.080
0.788
1.074 0.689
stretch 0.995
Jn(EXG)(ZPE) = 0.677 a
From ref 9.
the ethylene molecule and in model A of the activated complex. It was calculated by the expression12
where L is the eigenvector matrix, F is the force field matrix, and X, is the eigenvalue for normal mode i. Table IV illustrates the fractional contributions of the diagonal terms of the force field to the normal modes and clearly shows the extent of the participation of the HCX bending coordinates in the diff erent normal modes.
1623
1625
1.000
0.999
0.999
c:H2 deform
1342
1349 1444
1.000
1443
1.000
0.997 1.000
0.997 1.000
Discussion I n eq 1 the magnitude of the isotope effect is deter-,
C:Ho rock
1236 8113
1236 811
1.000 1.000
1.000 1.000
1.000
wag
94 3 949
956 974
1.002 1.003
0.986 0.972
a
102'7
1046 267
1.004
0,993
0.836
0.858
mined by the product of the three individual factors, ILIMI, EXC, and ZPE. It is evident from Table I that the values of the MXI factor are always greater than one and contribute toward a direct isotope effect regardless of the 6-X bond distance or the mass of X. This follows from the fact that addition reactions in general are accompanied by an increase in the moments of inertia and this increase is always smaller for ihe deuterated than protiated molecule. Inspection of Table I1 reveals that when no changes of the ethylene force constants are assumed in the transition from reagent to activated complex, that is when the reaction center preserves its original planar
stretch
cw2 m
twist
1.000 0.988
0,974 0.996 0,717
II(EXC)(ZPE) = 0.685 From ref 9.
The four normal modes assigned to the C-H stretching frlequencies were assumed to remain unchanged in the activated complex and are not included in Tables 11-117. Similarly, the normal mode assigned to the C 4 - X bend, because of its negligible sensitivity to the dcuterium substitution, is left out. Table I V gives the potential energy distribution of the normal modes among the internal coordinates in
(11) 0. P.St,rausz,H. E. Gunning, A. S.Denes, and I. G . Csizmadia, J . Amer. Chem. SOC.,in press; 0. E'. Strausz, R. K. Gosavi, A. S. Denes, and I. G . Csizmadia, Theor. Chim. Acta, 26, 376 (1972). (12) E. B. Wilson, J. C . Decius, and P. C. Cross, '"Molecular Vibrations," McGraw Hill, Kew York, N. Y . , 1955.
The Journal of Physical Chemistry, VoE. 76, A-0. 34, 1QYLy
I. SAFARIB AND 0. P. STRAUSZ
3616
Table IY: Potential Energy Distribution of the Diagonal Elements of the F Matrix in the Ethylene Molecule and In the Activated Complex (Model A)"
H"C2-C1
HLC2-C'
H"C'-H4
Hs-CS-HE
bend
bend
torsion
torsiou
stretch
1.5 (1.5)
1.5 (1.5)
CH2 deform
23.7 (23.2)
23.7 (23.2)
CHz
25 (26)
25 (26)
CHZ rock
25 (23)
25 (23)
CHZ rock
25 (24.5)
25 (24.5)
Normal modes
c-c
deform
CH2 wag
out of plane bend
50
(59)
CHz (6)
wag
CH:2 t,wist
50 (35)
(6)
out of plane bend
50 (37)
44
44
(35)
(49)
50 (35)
CHz twist 5
Values are given in per cent. Bracketed figures refer to the activated complex (model A).
structure, the frequencies related to the frequencies of the reactant ethylene molecule are hardly changed. However, the mere presence of X in the activated complex with only a very weak bond to the carbon atom creates a new, highly isotope-sensitive vibrational degree of freedom, the frequency of which determines the value of the product of the EXC and ZPE factors. Furthermore, it is apparent from Table IV, that the new normal mode contains a significant contribution from the CB2 tort4on coordinates in addition to that of the H-C-X bending coordinates, therefore its frequency must be significantly high even in the case of only a very weak C--X bond and correspondingly low H-C-X force constant. It should also he noted that a change in the mass of the reagent, radical (X) between 15 and 69 has no discernible effcct on the computed values of the isotopic rate ratio. RiIodel A with the described force field provides an 1.02. I n view of the inverse isotopa effect lc~/'kH large differonce from the experimental values, 1.071.14, it is unlikely that, as was proposed in the literaonly a negligible deviation of the transition state configuration from that of the reactant could generate an inverse isotope effect with the experimentally obtained magnitude Wolfsberg and Sterng attributed the secondary isotope effect to force eonstant changes in the reactant 2 :
The ,Journal of PWysiea.1 Chemistry, Vo1. 7%,N o . 94, 1979
when going over to the transition state. Our results illustrate that the creation of new vibrational degrees of freedom or the introduction of new force constants in the force field of the activated complex in thc process of addition can generate an isotope effect even in the case when the force constants of the reactant are kept unchanged. Table I11 illustrates the effect of altering the structure and the force field of the activated complex. As X approaches the C atom, the higher H-C-X bending force constant will increase some of the normal frequencies to different extents. It is seen from Table 111 that the frequencies of the activated complex related t o the out of plane wagging vibrations of the ethylene molecule somewhat increased, but contrary to Streitwieser's explanationY6this change alone cannot account for the observed inverse isotope effect. The magnitude of the isotope effect is principally determined again by the newly created twisting vibration in the activated complex. Model B with the described force field provides an inverse isotope effect, k ~ / k= ~1.04. If the effect of lowering the torsional force constant in the ethylene molecule is neglected, then k D / k H = 1.12. This gives an indication of the importance of the effect caused by the reduction of the rotational barrier of the terminal methylene group in the activated complex. The value JcD/lc~ = 1.12 calculated with mode1,B
CALCULATIQNS OF SECONDARY a-DEUTERIUM ISOTOPE EFFECTS without considering the reduction of the rotational barrier reproduces the experimentally obtained isotope effect. However, lowering of the torsional force constant in the ethylene molecule generates a direct isotope effect and decreases the calculated value for kD/kH. I n this case the experimental isotope effect can only be approached by further increase of the PY4 - X . and C--C-X bending force constants. The results obtained with model €3 show that the reproduction of the experimental value of k D / k H requires 13-C-16 bending force constant in the activated complex, about half as large as those in stable molecules. Although no physical theory exists at present which would relate the formation of a new bond to the accompanying changes in geometry and force constants of the reacting system, a gradual, probably linear, progression from initial t o final state can be assumed. Therefore the relatively large values for the H-C-X bexiding force constant in the activated state implies a Rtronger link between C and X and consequently a significant deviation from the planar ethylene structurc. It should be noted that the related normal modes of the ethylene molecule and the activated complcx, as nt ie apparent from 'Table IV, are not always identical. However, the formal resolution of the isotope effect according to individual frequencies shows clearly the profound influence of the change in the vibrational dcgrees of freedom on the isotope effect. It can be coricluded from our calculations that the isotope effect in addition reactions involving olefinic bonds i s the result of several factors acting in opposite directions : the moment of inertia increases, therefore the factor MMI is always larger than unity, and the BPE, owing to the increase in the force field by the creation of the new twisting vibration, always decreases. 'The values of the factor representing vibrational excitation are affected little by isotopic substitution. Conseauently the product of the three factors giving k I ) / k H , eq 1,wjll always lie close to unity by coincidence,
3617
but it would be conceptually incorrect to relate this fact to the structure of the transition state without a detailed analysis. Furthermore the present calculations clearly reveal that, contrary to the currently generally accepted notion, the main single source of secondary a-isotope effects in addition reactions involving olefinic bonds is the large increase in the force field caused by the creation of new, isotopically sensitive normal mode in the course of the reaction and not the relatively small force constant change which accompanies the rehybridiaation of the central carbon atom. This rule applies for all reactive systems irrespective of the mass or nature of the attacking reagent and whether the reaction is a simple addition or a true cycloaddition. Naturally it follows that all previous interpretations of secondary cu-deuterium isotope effects in addition reactions must be reassessed. I n the previous study on the S(3P) C2R4system the appropriate calculations could be carried out because of the availability of a satisfactory model for the transition state from the combination of accumulated experimental data and detailed ab initio molecular orbital calculations. At present there is no satisfactory physical model for interrelating force constant changes, and as their variations can be combined in many different ways to yield the same final results, agreement between calculated and experimental values generdly cannot be considered as proof of a particular activated complex. Therefore, in lieu of additional information about the structure of the activated complex it would be pointless to calculate the numerical values of force constants of the activated complexes. Konetheiess, our results illuminate the problem of the origin of secondary isotope effect within the framework of transition statc theory.
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Acknowledgment. The authors thank the National Research Council of Canada for continuing financial support.
The Journal of Physical Chemistry, Val. 7F, N o . 34, 1871
