A Calculation of the Energy Barriers Involved in the Isomerization

895

ENERGY BARRIERSIN THE ISOMERIZATION PROCESSES OF ETHYLENE

A Calculation of the Energy Barriers Involved in the Isomerization Processes of Ethylene in Its Excited and Ionized States by A. J. Lorquetl*

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Institut de Chimie, Universitt?de LiBge Sart-Tilman par Lihge I , Belgium (Received August 1, 1069)

The lowest singlet and triplet electronic states of the ethylene molecule and the lowest doublet and quartet electronic states of its positive ion have been studied by a semiempirical quantum chemical method. For each state, the energy of several nuclear configurationshas been calculated, in order to determine the preferred pathways of evolution. In the case of the triplet state, there are two low-energy processes. One of them is the cis-trans isomerization, and the second is a rearrangement process involving the ethylidene configuration (CHsCH). The results agree with the accepted views concerning the sensitized photochemistry of ethylene.

Introduction In a previous paper,’b the CzH4+ ion in its ground doublet state was studied by a semiempirical SCF calculation. It was shown that the equilibrium configuration of this ion was twisted, and that the potential function corresponding to the twisting motion had a double minimum and was remarkably flat. The exchange phenomenon occurring among hydrogen atoms (scrambling process) was also studied in the case of C2H4+,and it was concluded that this process took place through the asymmetrical configuration CHICH+. The purpose of this paper is to report a similar study for the mechanisms involved in the evolution of the first triplet state. A comparative study will be made of the properties of the lowest electronic states of singlet and triplet multiplicity of the ethylene molecule and of doublet and quartet multiplicity of the ethylene ion. The method of calculation has been described elsewhere.’b It is the CNDO/2z version of the selfconsistent field molecular orbital method, with slight semiempirical modifications.

Results The calculations have been made in the neighborhood of particular geometrical configurations, depicted in Figure 1, and conventionally designated as the “planar,” “twisted ,’’ ‘‘perpendicular,’’ “bridged,’ ’ “doublybridged,” “asymmetrical” (ethylidene), and “asymmetrically bridged” configurations. The latter configuration corresponds to the intermediate state postulated by Whalley3 in a study of the mercury-sensitized photochemistry of ethylene. Table I gives the energy of the most stable structure for each state and configuration, determined by varying every internuclear distance and angle, and measured from the ground state of the neutral molecule.

The method is found to overestimate the internuclear distances, but in a rather uniform way (20-25% increase). Several experimental results are available to enable US to determine the accuracy of the calculations. They are summarized in Table II.4-s We conclude that, in general, the method is capable of giving fairly reliable estimates of a t least qualitative significance. A . The Twisting Motion. The corresponding potential function has already’ been discussed in great detail for the doublet state of the ion. For the triplet state of the molecule and the quartet state of the ion, one notices from Table I, that this motion does not involve the angle of twist @ only, but is accompanied by an appreciable variation of internuclear distances and angles, especially in the case of the quartet state. Calculations were also made for intermediate values of the angle of twist (not reported here for the sake of brevity). They show that the potential function for the twisting motion is remarkably flat, especially in the neighborhood of the minimum. B. The Flapping Distortion. Walshe has suggested that in its first excited state, the CzH4 molecule should

(1) (a) Chercheur qualifih du Fonds National Belge de la Rech-

erche Scientifique; (b) A. J. Lorquet and J. C. Lorquet, J . Chem. Phys., 49, 4955 (1968). (2) J. A. Pople and G. A.Segal, ibid., 44,3289 (1966). (3) E.Whalley, Can. J. Chem., 35,565 (1957). (4) D. F.Evans, J . Chem. Soc., 1735 (1960). (5) J. E. Douglas, E. 8. Rabinovitoh, and F. S. Looney, J . Chem. Phys., 23,316 (1955). ( 6 ) D.P.Chong, and G. B. Kistiakowsky, J . Phys. Chem., 68, 1793 (1964). (7) A. J. Merer and L. Schoonveld, J. Chem. Phys., 48, 522 (1968). (8) W. C. Price and W. T. Tutfe, Proc. Roy. SOC.,A174, 207 (1940). (9) A. D.Walsh, J . Chem. Soc., 2325 (1963). Volume 74,Number 4 February 19, 1970

896

A. J. LORQUET

Table I: Energies of the Different Configurations of CzHa and CzHd' Configurations

Species

Neutral molecule, singlet state

Planar Perpendicular Bridged Doubly bridged Asymmetrical Asymmetrically bridged

Neutral molecule, triplet state

Planar Planar Perpendicular Bridged Doubly bridged Asymmetrical


Asymmetrically bridged

Ion, doublet state

Twisted Planar Perpendicular Bridged Doubly bridged Asymmetrical Asymmetrically bridged

Ion, quartet state

Planar Perpendicular Bridged Doubly bridged Asymmetrical L4symmetricallybridged

Geometrical parameters

Energy, eV

QCCH = 122";R(CH) = 1.371;R(CC) = 1.6lA;O = 0" XCCH = 1220;R(CH) = i.37A; qcc) = 1.67A;S = 900 R(CiHz) = 1.35A; R(CiHi) = 1.59 A; R(CC) = 1.81 A R(CC) = 1.71 A ; R(CH) = 1.71 b; R(CiH) = 1.36 A; R(CzH) = 1.38A; R(CC) = 1.70 A; CCzCiH = 180"; QCiCzH = 105' R(CC) = 1.85 A; R(CH1) = 1.58A; R(CHz) = 1.39 A; R(HiH4) = 2.52 A; CCzCiHz = 104' Vertical transition e = 00;CCCH = 1200; R(CH) = i.36A;~(cc) = 1.78A; 6 = 90"; KCCH = 121'; R(CH) E 1.37 A; R(CC) = 1.72A; R(CHi) = 1.57A; R(CHz) = 1.34A; R ( C C ) = 1.78A R(CC) = 1.661; R(CH) = 1.77b R(CC) = 1.70A; R(CiH) = 1.34A; R(C2H) = 1.36A; QCzCiH 1,SO'; CCiCzH = 107" R(CC) = 1.60A; R(CHi) = 2.191; R(CHz) = 1.38A; R(HiH4) = 0.96A; CCzCIHao= 133" CCCH = 12O0;R(CH)= 1.38A; R(CC) = 1.69.k; e = 37" CCCH = 1200; R(CH) = 1.3801; R(CC) = MOA; e = oo QCCH = 120°;oR(CH) = 1.38A; R(CC) = 1.68A; e = 90' E(CiHz) = 1.37 A; R(CiH1) = 1.781; R(CC) = 1.53A R(CC) = 1.7.&*R(CH) = 1.7L4 R(CC) = 1 . 6 9 i ; R(C1H) = 1.37A; R(C2H)= 1.39A; QCzCiH = 180"; CCiCZH = 104" R ( C C ) = 1.64A; R(CHi) = 1.66A; R(CH2) = 1.39A; R(H1HI) = 2.44 A; QCzClHz = 135" CCCH = 1280; R(CH) = 1.41 A; R(CC) = 1-72A; e = 0 0 QCCH = 1040; R(CH) = 1.38" A; R(CC) = i.sA;e = 900 R(CiHz) = 1.3tA; R(CiH1) = 1.59.k; R(CC) = l.86.k R(CC) = 1.624; R(CH) = 1.80A R(CC) = 1.80 A; R(CiH) = 1.41A; R(CzH) = 1.39 A; QCzCiH = 180'; QCiCzH = 94'. R ( C C ) = 1.87b;R(CHi) = 1.60b; R(CH2) = 1.38.&; R(HiH4) = 2.56 A; QCzCiHz = 128"

0 3.93 8.91 8.09 3.27

2.90 4.52 3.55 2.60 9.91 8.62 3.24 7.03 11-44 11-54 12.34 15.25 16.70 12.23

13.76 18,20 17.57 19.32 21.98 18.90 18.07

Table I1 : Comparison between Calculated and Experimental Energies Proceas

(1)Vertical singlet-triplet transition (2) Planar ground state to 90" twisted perpendicular ethylene in its singlet state (3) Planar ground state to lowest level of the triplet state (perpendicular) (4)Planar ground state to asymmetrical triplet state (ethylidene) ( 5 ) Twisting motion of the CIHI+ion in its doublet state (two barriers)b (6) Adiabatic ionization

---

Energy, eV-

Caled

Exptl

Ref

4.52 3.93

4.6 2.8"

4 5

2.6

2.3

6

3.24

3.0b

6

0.05 0.3 10.5

7 7 8

0.1 0.8 11.5

0 Activation energy of the cis-trans isomerization reaction involving no spin multiplicity change. energies and heats of hydrogenation.

have a pyramidal arrangement of the bonds about each C atom. We have investigated this possibility by calculating the energy of the system as a function of the angle between the molecular plane before distortion and the plane HlClHz after distortion, which we call 6. The Journal of Physical Chemistry

b

Estimated from bond dissociation

In the case of the singlet state of the molecule and the doublet state of the ion, in the planar and perpendicular configurations, the minimum of energy corresponds to a value of 6 equal to zero (no distortion takes place). I n the case of the triplet state, this kind of distortion stabilizes both the planar and the perpendicular con-

ENERGY BARRIERS IN

THE

897

ISOMERIZATION PROCESSES OF ETHYLENE HD

d

a

HZC=CDz

CD

e HDC=C.HD

(4)

The intermediate form is the "asymmetrically bridged" structure proposed by Whalley3 and represented in Figure lg. The energy difference between each of the possible intermediate forms and the lowest energy configuration was calculated by the CNDO method for the lowest state of each multiplicity. Again, the geometrical configurations giving the lowest energy were determined by varying all the internuclear distances and angles. The results are summarized in Table 111.

b

f

C

e HC

Table I11 : Energy Barriers for the Different Downloaded by FLORIDA ATLANTIC UNIV on September 1, 2015 | http://pubs.acs.org Publication Date: February 1, 1970 | doi: 10.1021/j100699a038

Rearrangement Mechanisms" ----Emol,

Figure 1. Geometrical configurations of ethylene and its ion: a, planar; b, perpendicular; c, twisted; d, bridged; e, doubly bridged; f, asymmetrical; and g, asymmetrically bridged.

figurations, but to a small extent. When the molecule is trans-bent, the largest gain in energy is equal to 0.13 eV for a value of 6 equal to 33"; when it is cis-bent, the gain is equal to 0.4 eV for a value of 6 of 42". When the perpendicular configuration is distorted, the maximum gain in energy is equal to 0.09 eV (6 = 31"). The latter distortion leads to the most stable configuration of the triplet ( E = 2.51 eV). Our calculations agree with Walsh's predictions, but they also show that the gain in energy obtained by the flapping distortion is rather small. C. The Scrambling Process. Four possible mechanisms for exchange phenomena among hydrogen atoms have been studied. H HzC=CDz

CD e HDC=CHD

, "HC

(1)

D The intermediate form is the "bridged" structure represented in Figure Id. H

PI HZC=CDZ

, "C

C

e HDCsCHD

(2)

D D The intermediate form is the "doubly bridged" structure represented in Figure le. H&=CD%

HC-CHD2

$ HDC=CHD

(3)

The intermediate form is the "asymmetrical" structure represented in Figure If.

Process

Singlet

(1)

8.9 8. I 3.3 2.9

(2) (3) (4)

eVTriplet

7.3 6.0 0.6 4.4

y---Eion, Doublet

3.8 5.3 0.8 2.3

eV--Quartet

1.8 4.4 1.3 0.5

a This table gives, for each state, the energy difference between each intermediate structure and the lowest energy configuration. In the case of the neutral molecule, the lowest energy configuration is, for the singlet state, the planar configuration, and, for the triplet state, the perpendicular one (the small gain in energy realized by the flapping motion has not been taken into account). I n the case of the ion, the lowest energy configuration is, for the quartet state the perpendicular configuration, and, for the doublet state, the twisted configuration.

Process 3 (via asymmetrical ethylidene) is therefore seen to be responsible for the rearrangement occurring in the ionic doublet and molecular triplet states, whereas in the case of the quartet state of the ion, a similar rearrangement process should probably occur through process 4. A question now arises concerning the exact significance of the energy differences given in Table 111-to what extent can these numbers be assimilated to activation energies? It is not evident that the energy variation along the reaction path will be of the type depicted in Figure 2a. A second possibility is depicted in Figure 2b; in that case, our calculations would indicate only a lower limit to the activation energies. It is very difficult to investigate this by means of calculations because the number of geometrical parameters to optimize for a distorted intermediate calculation is very large. We tried it, however, in a favorable case. One finds that the energy variation along the reaction path is uniform, as in Figure 2a. The calculation was made in the case of the doublet state of CzH4+ passing from the planar to the bridged configuration. It was supposed that during the whole reaction the molecule was planar and had a center of symmetry. Volume 74, Number 4 February 19,1970

A. J. LORQUET

898

Figure 2. Two possible variations of the energy along the reaction path.


The progression of the reaction was measured by the value of a, the angle between the C2C1 bond and the CIHl bond. The values of the other geometrical parameters were determined by successive approximations to obtain, for each value of a, the minimum value of the energy. The results are given in Table IV, where the

Table IV : Energy Variation along a Possible Reaction Path in the Case of the Doublet State of C2Haf Passing from the Planar to the Bridged Configuration R(CiHd, R(CIHI), R(CC), U

120 90 O 70” 64” 33’ O

B

A

A

A

E, eV

120’ 131” 161” 180’

1.38 1.37 1.37 1.37

1.38 1.40 1.53 1.78

1.70 1.73 1.62 1.53

11.54 13.05 14.75 15.25

energies are measured from the ground state of the neutral molecule, and p is the angle between the bonds CzCl and C I H ~ . In the other cases, a similar calculation is practically impossible. I n the intermediate steps, the symmetry of the molecule is very low. The number of possible reaction paths becomes very large, and the number of geometrical parameters to optimize for each point of a given path becomes prohibitively large. But this very difficulty makes it reasonable, although by no means certain, to generalize our previous result, and to argue that the number of possible different paths leading from A to B (Figure 2) is so large that it will always be possible to find at least one with no secondary maximum. The Sensitized Photochemistry of C2H4 The mechanism proposed by Callear and Cvetanovi6’0 for the mercury sensitized reaction of ethylene is generally accepted. According to this mechanism, CzH4 is first converted into a vibrationally excited triplet state molecule (designated as CzH**), which can undergo cis-trans isomerization by deactivation. This state (C2&*) cannot decompose directly to acetylene and hydrogen, but can isomerize to another triplet state (CzH4**) which can so decompose. The latter state (CeH4**)is also capable of undergoing extensive hydrogen rearrangement. For that reason, Callear and Cvetanovid suggested that the CzHh** state might have the ethylidene structure (CHGH; see Figure lf). The Journal of Physical Chemistrg

Our calculations agree with Callear and Cvetanovi6’s mechanism, since they show that there are two 10.w energy processes for the evolution of CzH4 in its triplet state: (1) cis-trans isomerization, with possible transformation into a quasi-free rotor, or (2) isomerization to triplet ethylidene. The other processes are characterized by much higher energy barriers (see Table 111). In particular, the state suggested by Whalley3 (asymmetrical bridge, Figure lg) as an alternative possibility for the structure of C2H4** is probably ruled out, From Table I, one sees that the most stable configuration of the triplet state is the perpendicular one (2.6 eV above the ground state). Next comes the asymmetrical ethylidene at about 3.2 eV, and then the planar configuration at about 3.5 eV above the ground state. The energy necessary to convert the molecule into a free rotor is thus, according to our calculations, slightly greater than that required t o induce the hydrogen migration. cis-trans isomerization of deuterated species is, however, always possible even if the energy of the sensitizer is reduced down to about 2.6 eV, because the formation of the perpendicular triplet by collision need not be a vertical one. Collisional deactivation of the perpendicular triplet should give either isomer with equal probability, at least in the case of ethylene-&, where there is no difference in energy between the two isomers in their ground state. On the other hand, hydrogen rearrangement, evidenced by the formation of anti-CHz=CD2 from CHD=CHD, is only possible if the sensitizer provides an amount of energy which we calculate as about 3.2 eV, This agrees with a recent experimental work,ll where it is stated “This leads us to speculate that when the energy transferred to the vibrationally excited triplet state from a sensitizer is lower than the energy of the lowest triplet of benzene (i.e,, 3.6 eV), it does not always cross over to E** even if it is not collisionally stabilized, but has a possibility to change t o another state (probably another vibrationally excited state) which undergoes only cis-trans isomerization.” Finally, there appears to be a certain probability for the scrambling process to occur in the singlet state, this time probably by process (4), i.e., the mechanism suggested by Whalley (ref 3 and last paragraph of ref 13). Remarks on the Ethylene Decomposition Reaction. It has been pointed out by Cundall12that the decomposition reaction

3C2H4(vibrationally excited)

---t

+

3CzH2 Hz

is endothermic even in the mercury photosensitization experiments. (10) A. B. Callear and R. J. Cvetanovih, J . Chem. Phys., 24, 873 (1956); R. J. Cvetanovib in “Progress in Reaction Kinetics,” Vol. 2,G.Porter, Ed., Pergamon Press Ltd., London, 1964. (11) 9. Hirokami and 8. Sato, Can. J. Chem., 45,3182 (1967). (12) R. B.Cundall, private communication.

899


EQUILIBRIUM CONSTANTS OF ASSOCIATING PROTEIN SYSTEMS Another possibility, however, would be that CzHz is formed in the singlet state, either from a spin-forbidden predissociation or from the vibrationally excited singlet ground state of ethylene. The second mechanism has been recently advocated by Hunxiker,13who has introduced a third intermediate state (C&f) in addition to the other two already postulated by Callear and Cvetanovi6.1° This new state would decompose into acetylene and hydrogen and would be produced from the triplet ethylidene by an htersystem crossing to some vibrationally excited singlet state. What would be the geometrical configuration of this new intermediate state, postulated by Hunxiker? According to our calculations there are three possibilities corresponding to the three possible low-energy configurations of CzH4 in its singlet ground state: (1)

the planar configuration, (2) the asymmetrical ethylidene, and (3) the asymmetrical bridge proposed by W h a l l e ~ . Concerning ~ the latter, an independent observation by Hunxiker in the isotopic scrambling might provide some possible evidence for the occurrence of the third possibility.

Acknowledgment. The author wishes to thank Dr. G. R. De Mar6 for suggesting the problem, and for a critical reading of the manuscript. She is also indebted to Professor R. B. Cundall for a fruitful correspondence. It is a pleasure to thank Professor L. D’Or for his interest in this work. The financial support of the Fonds de la Recherche Fondamentale Collective and of the Fonds National de la Recherche Scientifique of Belgium is gratefully acknowledged. (13) H. E. Hunaiker, J. Chem. Phys., 50, 1288 (1969).

Determination of the Equilibrium Constants of Associating Protein Systems. V.

Simplified Sedimentation Equilibrium

Boundary Analysis for Mixed Associations by P. W. Chun and S. J. Kim Department of Biochemistry, College of Medicine, University of Florida, Gainesville, Florida 5.9601 (Received August 86,1969)

+

A simplified procedure for the determination of equilibrium constants for mixed associations of the type i A j B F! AtBj or nA mB F! AtBj AhBl, where i -I- h = n, and j 1 = m, are described. The procedure is applied to a thermodynamically ideal situation, but its application in the anal) sis of reaction boundaries of any mixed association in a biological system is also considered. The equations outlined are based on concentration as a function of radial distance at sedimentation equilibrium.

+

+

In recent years, the theoretical treatment of assomB i? C has ciating protein systems of the type nA been based on the interpretation of data obtained by various physical techniques. 1--6 Nichol and Ogstons and Adams’ have described procedures for analyzing mixed associations of this type in an ideal system from sedimentation equilibrium boundary experiments. Adams, et d,* describe the equilibrium constants and nonideal term B,, evaluated from mixed associations using osmometric measurements

+

(@napp).

The principal drawback of these earlier procedures for determination of the apparent equilibrium constant and composition of the complex is the cumbersome manipulation of the data involved.

+

This communication describes a greatly simplified procedure for quantitative evaluation of the equilibrium constants of any mixed type of association in an ideal (1) R. F. Bteiner, Arch. Biochem. Bhphys., 49,71 (1954). (2) G. A. Gilbert, Proc. Roy. Soc., A250,377 (1959). (3) J. L. Bethune and G. Kegeles, J. Phys. Chem., 6 5 , 1755 (1961). (4) G. A. Gilbert and R. C. L1. Jenkins in “Ultracentrifugal Analysis in Theory and Experiment,” J. W. Williams, Ed., Academic Press, New York, N. Y., 1963, p 59. (5) L. W. Nichol, A. G. Ogston, and D. J. Winaor, Arch. Bwchem. Bwphys., 121, 517 (1967). (6) L. W. Nichol and A. G. Ogston, J.Phys. Chem., 69,4365 (1965). (7) E. T. Adams, Jr., New York Academy of Science Conference on Advances in Ultracentrifugal Analysis, Feb 15, 1968. (8) E. T. Adams, Jr., A. H. Pekar, D. A. Soucek, L. H. Tang, and G . Barlow, Biopolymers, 7 , 5 (1969). Volume 74,Number 4 February 19, 1970

A Calculation of the Energy Barriers Involved in the Isomerization

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