Theoretical analysis of the cycloaddition of ethylene

Chemical Physics 52 (1980) 151-163 @ North-Holland Publishing Company

THEORETICAL

ANALYSIS

OF THE CYCLOADDITION

E. KASSAB, E.M. EVLETH,

J.J. DANNENBERG*

OF ETHYLENE

and J.C. RAYEZ**

Cenne de Micanique OndularoireAppliqu&, 75019 Paris, France, and the Deparhnent of Chemistry, City Universityof New York, Hmrer College, New York, N-Y. 10021, USA Received 2 January 1980 Revised manuscript received 1 July 1980

The cycloaddition of ethylene is theoretically analyzed for portions of the excited singlet and triplet hypersurfaces using a combination of semi-empirical and intermediate level ab initio techniques. The semi-empirical UHF calculations on the addition of triplet ethylene and methyl radical to ethylene showed that these two reactions have comparable theoretical parameters, including activation energies, spin transfer and spin polarization at the transition state. For the ?.S+ZS excitedsingletstate surfaces, the results of both the ab initio and semi-empirical calculations are qualitatively the same and correspond to the classical ideas generated from orbital symmetry rules. At the ab initio level the results are quantitatively poor, partially due to the use of an intermediate level configuration interaction treatment. In particular, it was not possible to obtain other than a small fraction of the total estimated valence correlation energy in cyclobutane. The configuration interaction problem for both ab initio and semi-empirical calculations is discussed in detail.

1. Introduction

The goals of this article are two-fold. First, we will characterize the theoretical nature of the photodimerization of ethylene for portions of both the excited singlet and triplet hypersurfaces. Second, we will explore some of the methodological problems encountered in using semi-empirical and ab initio methods to elucidate photochemical reaction mechanisms. We are especially interested in developing semiempirical methods in order to investigate larger systems on which good quality ab initio calculations are presently financially impractical. The dimerization of ethylene and other related olefins to give substituted cyciobutanes (or the retroaction) is well studied both experimentally [l] and theoretically [2]. The orbital symmetry rules for the concerted 2S+2S process are a standard pedagogical exercise [3].

l

* City University of New York, Hunter College, USA. * Present address: Univeaiti de Bordeaux I, Talence, NATO Postodoctoral Fellow, 1975-76.

While existing theoretical work is in support of these rules, there are several important nuances with regard to the excited state surfaces. Firstly the triplet surfaces have not been explored. Secondly, recent work on the analogous Hq surface [4] indicates that a conceivable route for deactivation of the excited singlet state of the ethylene dimer could occur by a crossed (Dzd) approach of two ethylenes [4b]. Thirdly, the theoretical nature of the so-called doubly excited state at large ethylene-ethylene separations is not anticipated by the orbital symmetry rules [4]. In this article we will explore portions of the triplet surface which yield the triplet tetramethylene diradicd. For the singlet surface we will only explore planar face-to-face approaches having rectangular-trapezoidal carbon atom configurations. We will specifically treat the theoretical nature of the doubly excited state at both small and large ethyiene+thylene separations. Finally, we will discuss the problems encountered in using an intermediate level configuration interaction treatment.

1JI

E. Kassab et al. / 77worelicnfnnnlysis of!hc cycloaddirion

of erhpkne

and ethylene. This was done u&g & modification cf the Gaussian 70 m in which the size of the individual I d s-cl. 3G orbitals is related to the mrrelalion cw?gy

Tlncmajor portionsof the ab initio cal~~~~t~~n~ were performedusing the gaussian dique of Whitten and co-workers [S]. an over contractedvalence basis set, olg) for carbon and (5s/24 for hydromd with orbital exponents,

[SJ.Further studies using a ~~~~nt~r~d using the intermediate tevel CI W&Mrnffnt dl~ubeed below and lack of financial &a#b srrry oul a complete treatment. @fesU%ZPcalculutionswere performed oh dk# @l@#B#hell ground and open shell singlet &nd @ipfftt ctlnflgurutians. The CI treatment [6] 8%t9~ ir,irrlly symmetricstates was expanded

two parents,one being the original sh@llconflgmkm, the other being the nntigtration resulting from two promofcd from the HOMO to Is trentmentyields au after CI ground state, So, and the sonclted stnte, !P*. By symmetry

@QrMlUnn,the rlagtetand triplet states cal@&MrPj her@,87 ltndTI, have B1,, symmetries i&f the OJ~form of the dimsr, these being r&&d to r&Hasgeneratedby cxcitoa interHi8dttbof the mr* (“%J,,)stutes with the $$t%unrd 618IQ, Thu Cl cxpansious for these open sr&ll etztes WQ~Q performed around only one %L”i”parent cznllgurutian. All CI calculations WVrl@the result of the threshold terms [?i] of 5 x n with tho number of configurations 1 4OfJin each af the three separate ‘i%lpcl salcululions were performed by EreezMI oh@~OWVM 1 MO’s (essentially Is on C), the rNiano9 searchbeing done over the next

‘8, “Fhusthe highest 12 virtual orbitals %@~i-” not ttxaminad. Since the Ci calculations @@IF perfarmr;donly ORa portion of the &M arhitz\lswe decided to estimate the a%1 4%lencle currrelntiun energies of tiylobutane

PI. 2.2. Semi-empirical The method used is the previowfy moderately reparameterized CM)0 te&niq~e [2fl applied at the medium CI tevei @O-120~ CPI at the UHF level. An identical paratneter&&on was used which yielded, at the 60x60 level. an approximate enthalpy of -20 k&/mole [Zfj* (obs., -18) [lc] for the reaction of two ethylenes to give cyclobutane as well as reasoaabk geometries for these two molecules (e.g. CC distar.aes of 1.35 and 1.56 A, respe&vely). The semi-empirical CI treatment is to be contrasted with the ab initio one in that both the grauhd and excited singlet aud triplet Cl states were generated from the same set of CN00 SCF closed s!~ell molecular basis orbit&. In add&n, no z!ltomatic confguration selection procedure was used in the semi-empirical CI treatment. Important configurations were inch&d in the treatment as a result of a number of trial c2& culations in which the inadequzy of the Cl basis set was evident by disco&n&ties of the S** and Sf surfaces at the HOMN,UiMO inversion geometry (ca. 2.1 A separation between the two ethylenes). AIthougb the original calibration of the relative enthalpies of ethylene and cyclobutanewas done at the 60 x 60 CI level, it was found that the excited states were better treated at a larger CI level Thus, the So and S** states shown here were treated at the 102 CI level of which 88 were doubly excited including 26 four open shel9 amfigurations whose importance will be d&cussed later. The ST and T1 states were treated at the 75 md 84 CI level, respe-tively. At the time this study was done we had no available scheme for geometry optimization at the after-C1 level. We performed a partial point-by-point optimization for the S** state al t A misprinl in ref. [2fl

quotesthis value at 4 kczllmk

2.0 A ethylene-ethylene separation. The same was done for the ST state at 2.8 A. This afterCI optimization was done by varying only the CC distances and the CHI ffap angles. For the UHF triplet surface a more complete optimization was performed [g] using the same repatameterized CNDO method [2f]. However, exploratory calculations were aiso done using the unparameterized INDO as well as MINbO/S methods. These latter methods were rejected for reasons discussed below,

energies of different molecules [12] hes been established [13] and largely corrected at the MCSCF level [14]. Likewise, we have demonstrated that our reparameterized CNDQ CI method yields adequate appearing ground and excited state bond rupture surfaces [2f]. It remains to be shown by comparative calculations, however, whether or not a CNDO CI method can generally mimic the excited state surface features generated by an ab initio calculation of similar basis size. Thus, part of what .-Jill be explored here is a comparison of such calculations and problems encountered.

3. Reaulls nnlddiscussion 3.2. The ethylene-ethylene triplet dimerizatiotl 3.1. Gerteral methodological onsideratiorrs It is our general view that useful inexpensive information on excited state behavior can be obtained using semi-empirical methods [2f, 91. The disrepute of such methods lies mainly with their quantitative uncertainties and seemingly ever changing parameterization schemes. It 1s probably a general opinion that for small molecular systems with a large basis set, large CI cnlculation can be trusted for a surface calculation involving some changes in relative correlstion energies along that surface [lo]. For huge molecular systems, financial or technical limitations impose smeller basis sets such ih:lt a lorgc CI treatment may be difficult or presently impossible and the resulting surfaces poor [IO]. Since both ab initio and semi-empirical calculntions using similar size basis sets will carry the same symmetry information, semi-empi::ical CI methods still have the potential of giving useful information on large systems, In this case the advantages of the small basis set semiempirical CI method over its ab initio counterpart lies in its variable parameterization. Th,us, the CNDQ/S method [l 11, will still give be.ter estimates of transition energies For a large system tharl will a small basis set ab initio method. Neither the CNDCI/S nor the original CNDQ/INDO methods were originally parameterized with the intention of doing s.uface calculations. The main reason for their poor estimate of the relative ground state

Available thermodynamic, spectroscopic and theoretical data indicate that the reaction of twisted triplet ethylene (3E90-, CC = 1.48 a [IS]) with ground state ethyfene (‘l$) to give the triplet tetra-methylene diradicai (“TMDR) is about 19 kcal/mole exothermic: ‘Eob+ 3Ego.= 3TMDR,

AH = - 19 kcal/mole, (l!

‘I$,- -I-‘I& = cyclobutaae, AH = - 18 kcal/moIe [ Ic], cyclobutane

(:!‘;

= ‘TMDR,

AH =53 kcal/mofe [ICI, *Eoo+ hv = ‘Euom, AH = 64 kcal;‘mole,

13) (1)

This estimate (1) is obtained by combining reactions (2), (3), and assuming that ‘TMDR and ‘TMDR are nearly isoenergetic. Only in reaction (2) do we have a true experimental value. For reaction (3) we assume that, the activation energy for the thermal decomposition of cycfobutane is identical to the enthalpy change generating ‘TMDR even though it has been theoretically indicated [Zc] that there may be several ‘TMDR intermediates which are several kcal/mole lower in energy. This would make the value shown for (1) slightly more negative. Finaflv, the value shmvn for reaction (4) is theoretical [15a. b]. Llowevcr. theoretically ‘Ego0and ‘Ew are nearly isoenergetic and thus the energy of reaction (2)

E. Kassab et al. 1 theoretical analysis of the cycloadditiin of ethylene

154

will approach the known activation energy for thermal cis-trans isomerization of ethylene [lc]. Even though triplet sensitized photodimerizations of small ring oIefins have been commonly observed [le-j] reaction (1) has never been fully demonstrated to occur with ethylene .of simple acyclic olefins. This observation WI be conveniently rationalized [lg] by examining the kinetic expression for the disappearance of triplet olefin (3A) by competitive unimolecular intersystem crossing 3A%1A,,

(5)

and bimoIecular dimerization 3A c ‘A,, 2

3DR_

(6)

The fates of the generated diradical, ‘DR+ ‘DR, would only be important if for some reason the regeneration of starting material from ‘DR was highly favored only in the case of acyclic olefins. if we assume that only reactions (5) and (6) dominate in both types of olefins, then: d(3A)/dt = -!c~(~A) - kX3A)(‘A,J.

(7)

The trajectory calculations of Warshel and Karplus 1161, indicate that ki, in ethylene is strongly dependent on the 3AS(l,1A90- energy gap, a result similar to the known experimental and theoretical energy gap dependence for triplet-ground state intersystem crossing in aromatics [17]. The near degeneracy of the tripIet and ground state surfaces in olefins occurs onIy near the twisted 90” configuration of the p-orbitals comprising the z-bond. While the triplet states of small ring olefins will undergo some relaxation, it is axiomatic that the T-S energy gaps in relaxed cyclic olefin triplets will be larger than for acyclic triplets. Thus, it can be argued that it is the variation of kk with olefin structure which controls the partitioning of reactions (5) and (6). There is an additional nuance to the arguments posed above. If the values for kd are less than diffusion controlled ones! this implies enthalpyas well as entropy of activation effects

in reactions of the same type as (1). It is commonly assumed that triplet addition reactions are radical-like in character. If so, one can anticipate radical-like kinetic parameters. Our goal will be to estimate activation energy and enthalpy for reaction (1) as well as to attempt to characterize the radical-like nature of the reaction pathway using geometry and spin transferspin polarization criteria_ We will also investigate the possibie differences in triplet sensitized photodimerization cyclic and acyclic olefins. We will only report in detail on the computed optimized triplet reaction path for reaction (1) generated using our reparameterized CNDOUHF method [2f]. It was initially determined that an unreparameterized INDO-UHF method gave a hopelessly false enthalpy for reaction (1) (ca. -100 kcal/moIe, versus -19 estimated) as well as intuitively false geometries for 3TMDR and intermediate structures. While the MINDOi3 half-electron method gave a reasonable estimate of this enthalpy (-31 kcal/moIe), the optimized C-C geometry for 3E90n(1.36 A) was very far from the best ab initio value (1.48) [15]. Likewise, the MIND0/3 optimized geometry for trans3TMDR gave 1.44 8, for the -CHz-CHz distance, somewhat far from what one would expect from a C-C sp3-sp2 hybrid (1.52) [IS]+. Our own CNDO-UHF reparameterization yielded -27 kcal/mole for the enthalpy of reaction (l), and 1.49 and 1.52 for the C-C distances in 3Egc. and 3TMDR. Since the same CNDO reparameterization is used in our CI calculations discussed in the next section we decided to retain the same overall parameterization for both the UHF triplet and RHF-CI singlet surface in spite of the fact, as will be shown, that the final computed activation energy for reaction (1) was not satisfactory. It should be stressed, however, that from a general methodological point of view the use of a UHF single determinate method to compute a surface of a bimolecular reaction between what are, at a dissociation limit, a closed and open shell species, will give rise to a relative corf See also ref. [k].

E. Kassab et al. /

Theoreticalanalysisof the cycloodditionof ethylene

relation energy error [19]. In any case, the following energetic analysis is given using the reparameterized CNDO-UHF approximation. First, in order to determine the energetics of the reaction of nearly rigid smali ring triplets we computed the following UHF geometry optimized reactions using our modified CNDO parameterization: 3Eoa(CC= 1.35) = 3Erp(CC = 1.54), AH = -28 kcal/mole, 3Eoe(CC= 1.54) = 3E90.(CC= 1.49), M = -16 kcal/mole.

(8) (9)

Reactions (8) and (9) have previously been estimated by Baird and co-workers 1201 at -32 and -16 kcal/mole, respectively, with virtually the same optimized CC distance. As a model calculation for the cyclopentene triplet, we performed geometry optimizations on ethylene in which only two cis-hydrogens and the CC distance were allowed to vary and found an essentialIy planar structure with the same CC distance and energy as shown for reaction (8). Thus, in a certain sense, small ring cyclic olefin triplets contain about 16 kcal/mole excess energy as compared to relaxed acyclic olefin triplets. The geometry optimized CNDO-UHF minimum energy pathway for the reaction of 3Eg0. with the ethylene ground state is shown in fig. 1. For comparison purposes, the reaction of the methyl radical with ethylene is shown in the same figure. The critical coordinate is the CC bond distance between reacting carbon centers, this being sufficiently large at the transition state (ca. 2.5 A) that little activation energy difference was found for rotational variants about the forming CC bond. The optimized geometry of the tfansition state for reaction (l), shown in fig. 2, shows little geometry change in the reacting moieties as compared with their isolated structures. The possibly more critical measure of progress along the reaction coordinate is the spin density. In both reactions there is virtually no spin transfer at the transition state. Both the geometry and spin density transfer criteria are consistent with the inter-

155

pretation that the transition state is more reactant-like than product-like. On the other hand,

while little spin transfer has occurred, the accepting carbon atom in both reactions exhibits a large negative spin density, a polarization which occurs well before the transition state. Even though an unprojected UHF calculation will overemphasize such spin polarization, what is observed demonstrates the principle that a negative spin density at the receiving carbon atom in radical reactions is a required precondition for bond formation [22]. What we had not anticipated is that such spin polarization would occur at distances very much larger than that occurring for the threshold for spin transfer. In any case, the theoretical profiles for both radical and triplet addition to ethylene are similar, giving theoretical support to the idea [22] that some triplet state reactions are radicallike in character. As discussed below, the main negative feature of the CNDO-UHF calculations presented is that they give unreasonably high activation energies. However, we decided to use our CNDO-UHF calcuiation of the methyl radicalethylene reaction to estimate the probable activation energy of reaction (1). Reaction (1) has a computed activation energy of 28 kcaljmole while that of a simulated cyclopentene triplet-cyclopentene is 21 kcal/mole. This latter value is comparable with our own computed value, 19 kcal/mole, for the methyl radical-ethylene reaction which is, in turn, about a factor of two, too large (ohs., 8 kcal/mole) [23]. Thus, it can be argued that the olefin tripiet-olefin reaction should have an activation energy comparable to normal radical addition reactions. Therefore, based on this comparative method we predict that the reaction of cyclic olefin triplets with olefins should have activation energies in a region of 8 kcal/mole. Acyclic olefin triplets may have activation energies several kcaljmole higher than this value. Thus, it is predicted that the low quantum yields of acyclic triplet sensitized photodimerizations are due to a combined high value for kec and lower than diffusion controlled rate for kd. However, in cyclic olefins ka should

CARBQN-CARBON

BOND

DISTANCE,

ANGSTROMS

Flfl. I. Rep~rnmcterizizd [2f] CNDO4JHF calculations of the addition of triplet ethylene to ethyk~ and m&,4 r&k%! BD cthylct~c. Shown is the minimum geometry optimized pathway for an in-plane carbon amm co&go&o% wi~-irhCfM &T%B~ kept constnnl al 1.08 A. Also shown are the spin transfer from the radical or triplet species to e~I~@oe and the mzgz~~ spz~ dcnuily n1 the occepling carbon atom along the reaction coordinate.

.. nlso be lower than diffusion controlled rate. l’hus, in a renl sense, ir is the variation in the kilir values which differentiate the triplet sensitized phatodimerization kinetics of acycIic and cyclic olcfins. Sin&r effects should occur in cyclic and acyclic polyenes. It is known that the qunntum yields for sensitized photodimerization OFcyclohexadiene (ca. 1) are much higher than for butadiene (O.Ol) [22]. On the other nand, \ve have found virtually no reported experimcntul values for activation energies for triplet nddition reaction aside from a recent value for tht! reaction of triplet trimethylenemethane with a substituted olefir~ (6 kcal/mole) [24]. 3.3 LVe cthyltme-eth ylene excited singlet surfaces 3.3. I. Qttalitative aspects The essential details of the calculations presented here are shown in figs. 3 and 4 for the semi-empirical and ab initio calculations,

respectively. The detailed energeti of the latter calculations are shown in tablie P. I%e geometry choices for the ab in&o ~~~~~ at the intermediate geometries (ca.!cuLat&~, B-5 table 1 and fig. 2) were bzsed cm S&ese1?3empirical results, oost amsi&rat&ms preventing us from doing a surface sear& fcr the 9* minimum. In both the semi-empfal a& & initio czlculatians the minimum d timeS” 51ifface is in the 2.0-2.2 A regk~~ Qualitatively, the ab initio ay.xdse&e calculations show the same be portions af the H, surface wea Michl and co-workers 13 possible photochemical faces has already been these workers. We will dli ia detail s~mr:d the complications invoived in treatig a very much larger system, especially in wbzt mzmncr OUTcalcul2tions are diEerentia@d &om mrE+ calculations on the same system.

E. Kmab

er al. f Th@orelfcal at&is

Fig 2. The geomeby of the uptiMid transition state of the calculation shown in fig. 1. A&J shown are the individual atomic spin deMies as well I the total amount of spin density tidefred of retained on each ethylene udit.

Both the semi-empirical and ao initio ca!culatiors give essentially the same qualitative infarmation, namely that there are two excited states Sf and S** of the ethylene dimer which

oftlthz cyclnaddirion

ofethylene

157

are involved in the photodimerizadon process. Initially, ar large ethylene-ethylene separations, tbe ST surface is lower or nearly isoenergetic to the S** state, but dt the HOMQ-LUMP avoided state crossing region near 2.0-i-2 8, ethylene-ethylene separatian the S** state becomes lower in energy. A5 seen in table 2, at short ethjrkne-ethylene distances, the configurational compasitinns of the SOand s”” states are as expected from orbitel symmetry rules. They both consist mainly of plus and minus combinations of two major configurations, one the closed shell SCF solution (HOMO doubly occupied) and the other the doubly excited configuration (LUMO doubly occupied). The relative weights of these two configurations change radically for the SOand S** states over a Iairly narrow geometry region of

significence

singlet excited stare, S’, and dauhty excited state, S**.

E. Kasab eta!. / ?heoreticaI analysis of the cycloaddition of ethylene

158 I

I

simple, it becomes largely monoconfigurational at the cyclobutane or two ethylene limit. The configurational behavior of the S** state is much more complicated. It does not evolve towards a S** state of monomeric ethylene as might be indicated by a 2 ~2 CI [2b] or a state correlation diagram. As in the case of I& the S** state correlates with two ethylene triplets [4,26]f. This is shown in table 2 where the S** state is shown to be neither principally a closed shell doubly excited state nor a two open shell monoexcited state but a four open shell doubly excited state at 3.5 A. The calculation shown in table 2 is done under conditions where the ethylene dimer is in a trapezoidal geometry where the MO’s are Iocalized on either one ethylene or the other at large ethylene-ethylene separations. If the dimer is calctdated under DZh symmetry, a set of delocalized MO’s is obtained which have a correspondance with those anticipated from orbital symmetry rules. Unfortunately, the configurational composition of the S”* state at large ethylene-ethylene separations depends on whether one is working with localized or delocalized MO’s In the case of

Fig. 4. Ab initio CI calculation of the cycloaddition of ethy!ene to give cylobutane. See table 1 for geometry detaiIs. Caicularion used an after contraction 40 orbital basis set for cyclobutane plus CI at about the 400x400 level using a configuration selection threshold of 5x IOm1hartree. Energies for the e:hylene (e) dimer limit (00) are twice the energies shown in table 1.

than it is usually. Likewise, this surface instability will permit the formulation of diabatic functions interconnecting the SOand S**.surfaces [257. More simply stated, this geometry region is sutliciently theoretically special to warrant the speculation that it is where the internal conversion from S** and So occurs [4d]. A detailed discussion is necessary concerning the correlation of the S** state at large ethylene-ethylene separations [26]. The configurational behavior (table 2) of the So state is

? Note that this point is overlooked in the correlation diagrams proposed in related types of photocycloaddition reactions: see for instance ref. [27].

Table 1 AS initio SCF-CI calculations on ethylene-ethylene Cal.

.9

R,

RL

R,

State

SO

1 2 3 4 5 6 7

IS 1.41 13 1.40 11 1.39 9 1.38 3.8 134 ethylene” cyclobutaneb’

1.415 1.425 I.435 1.440 1.500

1.90 2.00 2.10 2.20 335

S**

S”

T*

SCF

CI

CI

SCF

-155.562 -155.640 -155.700 -155.747 -156.875 -17.952 -155.961

-155.955 -155.936 -155.931 -155.945 -155.078 -78.079 -156.029

-155.745 -155.809 -155.838 -155.825 -155.733 -77.477

-155.562 -155.767 -155.639 -155.789 -155.645 -155.792 -155.644 - 155.787 nonconvergent -77.526 -77.672

CI

SCF

CI

-155.736 -155.757 -155.766 -155.771 -155.723 -77.808

-155.866 -155.886 -155.893 -155.889 -155.908 -77.852

a)Experimental geometry. ‘) Planar geometry as found in 6-31G* calcuiation of Cremer [32]. Other geometries were taken from semi-empirical calculations, geometrical parameters not shown were not changed, CH kept constant at 1.09, HCH bond angle at 112’. See fig. 1 for definitions of S, R,. R2 and RO_

E. Kassab eta!. / Theoreticalanalysis of the cycloaddition of ethylene Table 2 Leading configurational

terms in the ab initio calculations

Config.

Ri’

coeKb’

=

(HOMO)’

1.9

So s** So .S** So !Y* SO .S** So S**

= = = = = = = = =

0.215 0.900 0.379 0.846 0.662 0.539 0.842 0.397 0.929 -0.048

2.0 2.1 2.2 3.5

(LUMO)’ + -I+ c -

0.919 0.200 0.860 0.359 0.665 0.639 0.420 0.807 0.151 0.213=j

” AngtrBm units. ‘) Either double occupation of HOMO or LUMO. ” For this distance the major configuration is a four open shell having a value of 0.771. this configuration is related IO two ethylene triplets, see text. d’ For both the S* and T states at each distance, the major configuration (0.9.5-0.96) has a single occupation for the HOMO and LUMO.

between these five configurations and two triplet ethylenes at large ethylene-ethylene separations is difficult to see. Our general conclusion is that in this and similar types of calculations it is best to impose a slight symmetry breaking on the system is order to obtain localized orbitals at large separations. With respect to the ab initio calculations the configurational behavior of the SF and T1 states is simple, they are both largely purely HOMO+ LUMO open shell states (over 90% monoconfigurational). With regard to the semiempirical calculations the lowest energy S* state (‘Bz in C,,) has Z-U* not Z-T* character at large ethylene-ethylene separations but undergoes change in configurational character at small separations and becomes JXT*. This is an artifact of the CNDO method even under our parameterization.

3.3.2. Quantitative aspects-the localized orbitals (table 2) the double triplet character of the proper (S2 = 0) four open shell configuration is hidden within the polydeterminatal structure shown below:

where the indices 1 and 2 refer to orbitals largely localized on either ethylene 1 or 2. The energy of this configuration at large ethyleneethylene distances tends to become bitriplet in character because only the central two determinates contribute to the orbital-orbital exchange term K. Likewise these central determinates are triplet in character because one has either both (Yor fl spins in the same molecule. This behavior is also found in the semi-empirical calculations performed using localized orbitals. However, using del6calized orbitals the S**state becomes polyconfigurational at large distances, i.e. HOlMO’+ LUM02; HOMO’+ (LUMO + 1)‘; (HOMO l)‘+ LUM02; (HOMO - 1)2-+ (LUMO -I-1)’ together with the four-bpen-shell determinate shown above. The conceptual relationship

1.59

correlation

energy problem

With regard to the computed ground state energetics of the cycloaddition of ethylene, there are a number of previous calculations at the ab initio SCF [2b, 2c] SCF-small CI [2b, 2~1 as well as ihe semi-empirical level [2a, 2d, 2f2i]. At the SCF level, the ground state thermodynamics are greatly in error using a small ST0 [2b] or STOJG [2c] basis set (talc. ca. -80 kcal/mole, obs., -18) [lc]. Essentially experimental values are obtained from 4-31G and 6-31G* calculations [28]. With regard to the STO-calculations [2b] a simple 2 ~2 CI showed essentially no energy lowering for cyclobutane but about 0.08 au (ca. 50 kcal/mole) for two ethylenes. Similarly a srpall CI treatment [2b] at the STO-3G level shows a much greater energy lowering for ethylene than for cyclobutane [2c]. These small C&mall basis set treatments essentially improve the apparent agreement between the calculated (after CI) and observed enthalpies of the reaction by lowering the energy of ethylene more than cyclobutane. This implies that the total valence correlation energy of two ethylenes is much different than

for cyclobutane which, as we wid show, is not the case. Our own calculation at the SCF level yielded a computed enthalpy for this ethylene cyeloaddition reaction of -44 kcaf/mole (exptl. -!,R) which while better than the STO-3G-ST0 cefculationa quoted above is worse than a 41 I G or 6-31G” estimate. However, imposition af Cf at about the 400x400 level makes this value in even worse*(ca. +bO kcal/mole) ngrcrmcnt with the experimental value than at the SCF level! The possi,bfe sohnxon of imptovbrg rho SCF treatment using a larger basis set will not improve the above agreement except at the SCb Icvel. The problem resides in the CI technique used. An analysis of this problem requires some estimate of the correlation energies of both ethylene and cyclobutane in order to find cut how much of this energy is not being cnlculntcd. Shown in tnble 3 are the estimated correkition cncrgies for each localized carbon 1s ?orc, rmd the CH and CC valence orbitals for koth ethylene and cyclobutane as obtained from a correlative equation [7] relating the SKI-3Ci localized orbital size to the correlation energy. In the RISCof ethylene, the computed total corrclntion energy, OS4 au, is sufficiently close 10 the estimated vult~c (0.522) 17,291 that some

l’ilhk I ~‘~~~IcIIII~B~ energy __Ix_

-.-_..

h~
cyclulullrrllc

Donrl or orbital type lacnlircd In Cti CT

Is

C.~~f CC

__,“_” .,...*., _-_

AimatCs

___

fat ethylene

and cyclobutane

((rllr’l$))“* “R,

Correlation energya’ (au)

(1.316 0.1040 1.516 0.0560 1.6X 0.0546 correlation energy, core * 0.207 vnlctke shell * 0.333 total * 0.540 0.317 0.1033 I.515 0.0560 1.544 0.0567 corrclntion cnurgy, core = 0.413 velencr shell = 0.671 total = 1.084

” I’idculuted from the equation, Emrr = -0.06593 IIIR,, ” ‘V’h), see textnnd ref. [7].

x

confidence can be placed in ~dd~t~o~~ par& tioning of this energy into core and valence terms. Our computed valence correlation energies of ethylene (0.333) and cyclobutane (8.671 j are such that there is essentizify no great carrelation energy difference for the reactioraof two ethylenes to give cyclobutane. Thus, if we were onfy interested in the enthafpy of this reaction, we would not wish to go beyond a large basis set SCF estimate. With regard to a very large basis set-nearly complete CI treatment of the ground state of etiyfene, the valence correlation energy actmlly obtained has evolved over the years from intermedir?e values of about 0.2 au to the most recently co;.rputed value of about 0.33 au (0.36 estimated] l30]. Our own middle level CZ treatment yielded only 0.13 au for ethylene and 0.068 for cyclabutane. Especially in the case of cyclobutane the Cl treatment has only obtained about 10% of the total valence correlation energy. We know from the above discussion that the calculations shown in fig. 4 and table 1 .re faultsd because of an imbalance in the estimated relative correlation energies of ~~ctobutane and two ethylenes. The reason for this imbalance is that the CI ~afcufation estimates between cyclobutane and ethylene are not size consistent [31]. Since the CI space of ethy;;ne is smaller than cyclobutane the same configurational selection criteria (5 X lO-4 au) will yield more correlation energy in the case of ethyfene thhn for cyclobutane. Thus, from a size consistent stand point, only calculations l-5.7 shown in table I and fig. 4 are comparable. Does this mean that we can regard that portion of fig. 4 (and table 1) as more “believable” if one rejects the consideration of the two ethylene limit? Two things argue against this ahernate pofie. First, since only 10% of the valence correlation energy is estimated in the case of the grouno state of cyclobutane there remains a large margin of error. As one proceeds afong the surface some of the orbit& will fncrease in energy and states witi approach one another. fi is likely that the same configuration selection threshold energy wit1 produce a much higher percentage of the vafence correlation energy

.G. Karsnb

et al.

/ Theoreiicqt

analpis

along certain portions of the surface than along others in addition to that obtained from a mere HOMO-LUMO indwcd avoided crossing behveen SOand S*” (i,e. a ‘2x 2 CI). In the case treated here, the computed barrier for the concerted 2S+2S transition state for the ground State surface decomposition of cyclobutane to two ethylenes is less than 60 kcal/mole. This is less than the observed activation energy (+63 kcal/mole) [lc] for a process which is probably biradical-like in character [2c]. Our conclusion is that an intermediate level CI treatment cannot give adequate results on this surface regardlessof the irrikd ab initib basis set. With regard to the CNDO-CI calculations it must be pointed out that a similar size consistency problem exists with regard to the CI This can be demonstrated by examining fig. 3. Originally the overall thermodynamics of the cyclobutane-two-ethylene reaction was calibrated at the 60x60 CI level by individual calculations on cyslobutane and a sblgle ethylene to give the experiniental value of about 20 kcal/mole [2f]. In the surface actually generated in fig. 3 the same configurational composit,an was maintained along a soxface which begins with cyclobutane and terminates at 4.5 A with an ethyb;e dimer. The So surface in fig. 3 does not asympotitically approach the 20 kcal/mole calibrated value at large separations. This results from the fact that as with the ab initio CI calculation it is easier to compute a greater percentage of the correlation energy when the system is smaller (e.g. ethylene) than larger (ethylene dimet) using the same number of configurations. However, unlike the ab initio calculations, a semi-empirical technique can be recalibrated under size consistent conditions (i.e. cyclobutane and ethylene dimer at large separations). Likewise, since the semiempirical-after-Cl results are parameter dependent, a completc CI is not necessary as long as the major correlatively important configurations are included. In the case of the S** surface at large ethylene-ethylene sepatations the semi-empirical CI treatment must include doubly excited four-open-shell configurations. In principle, the inclusion of

of the cycloaddition

ofethykne

161

such configurations would involve a large CX treatment requiring a Parge search ta obtain rhe important ones* In fact, the important configuratiuns can be detelm,ined by a few preliminary calculations at various points along ?he reaction surface. Ir this mtmner, the semiempirical Cf treatment can be kept wlthin the level 100 x 100. However, it must be pointed out that the amount of semi-empirical “carrel&ion” energy will also depend on the size of the Cl treatment. At the 60 x 60 level this energy was found to be 0.045 au for the So stntc of ethylene, increasing to 0,087 au ai the 1lllx 110 level, the latter containing all totally symmetric mono- and dr;uble+xcitations. The latter energy obviously has no relationship to lhc above discussed estimated valeilce correlation energy (table 2) of 0.33 au. Likewise, a semiempirical potential eaergy curve calibrated with one set of configurations Will be out of calibr;ltion under another set. With regard to I~.:: problem treated here, $1

comparison of the semi-empirical and ab initio stirfeces (figs. 3 and 4j shows that there is a VISUJresemblance in the two computations ~II the region of avoided crossing at 2.0-2.2 A ill: well as along the surface between 2.0 to 4.5 h. The semi-empirical calculation gives the same correlative information as does the ab initio one. Both sets of calculations, however, Jo Ilot give us an accurate idea as to the height of the So and S*‘kstates in the region of thy HOMOLUMQ crossing,

For the triplet diradical surface reparameter&ted CNbO-UHF calculations indicate that the reaction of olefin triplets with olefins should involve activation energies similar to I-adical-olefin reactions. The theoretical profi!t.s

of both types of reaction are similar, as measured by spin transfer and spin polarizotiotl ;rt the transition state as well as the general :!ppearance of the reaction diagtan?. A comparison of ab initio sod semi-empilic“l! I :I calculations for the 2S+ 2S cyctoaddition

162

E. Kamb et al. / Theoretical analysis of the cycloaddition of ethylene

surfaces shows similar qualitative behavior. We demonstrate that the configurational behavior of the S** state at large ethylene+zthylene distaxes is unlike that prediCted by simple orbital symmetry rules. This state correlates with two triplets of ethylene. FinAlly, we show -that the system cannot be treated using ab initio techniques at the intermediate CI level. A separate

estimate of the valence correlation energy of cyclobutane shows that our calculations only obtained about 10% of that energy using a configuration

selection of 5 X low4 au.

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