theoretical analysis of the cycloaddition of ethylene

8/9/2019 Theoretical Analysis of the Cycloaddition of Ethylene

1/13

Chemical Physics 52 (1980) 151-163

@ North-Holland Publishing Company

THEORETICAL ANALYSIS OF THE CYCLOADDITION OF ETHYLENE

E. KASSAB, E.M. EVLETH, J.J. DANNENBERG* and J.C. RAYEZ**

Cenne de M icanique Ondul aroireApp l i qu& , 75019 Pari s, France,

and t he Deparhnent of Chemi stry, Cit y Uni versit y f New York, Hmrer Coll ege,

New York , N- Y. 10021, U SA

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 excited ingletstate 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

t o

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 hypersur-

faces. Second, we will explore some of the

methodological problems encountered in using

semi-empirical and ab initio methods to eluci-

date photochemical reaction mechanisms. We

are especially interested in developing semi-

empirical methods in order to investigate larger

systems on which good quality ab initio cal-

culations 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].

*

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 separa-

tions 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 inter-

action treatment.


2/13

1JI

E. Kassab et al. / 77worelicn f

nnlysis of hc cycloaddirion of erhpkne

Tlnc major portions of the ab initio cal-

~~~~t~~n~ ere performedusing the gaussian

dique of Whitten and co-workers [S].

an over contractedvalence basisset,

olg) for carbon and (5s/24 for hydro-

md 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%ZP alculutionswere performed oh

dk# @l@#B#hell ground and open shell singlet

&nd @ipfftt ctlnflgurutians. The CI treatment [6]

8% 9~ ir,irrlly

symmetric tates was expanded

two parents,one being the original

sh@ll onflgmkm,

the other

being the

nntigtration

resulting from two

promofcd from the HOMO to

Is trentmentyields au after CI

ground state, So, and the so-

nclted stnte, P*. By

symmetry

@QrMlUnn,he rlagtetand triplet states cal-

@&MrPjer@, 7 ltndTI,

have

B1,, symmetries

i&f the OJ~ orm of the dimsr, these being

r&&d to r&Has enerated by cxcitoa inter-

Hi8dttbof the mr* (%J,,) tutes with the

$$t%unrd

18IQ, Thu Cl

cxpansious for these open

sr&ll etztes

WQ~Q

performed around only one

%Li arent cznllgurutian. All CI calculations

WVl@ he result of the threshold terms [?i] of 5 x

n with tho number of configurations

1 4OfJ

n each

af the

three

separate

i%lpcl salcululions were performed by

Ereez

M I oh@

OWVM 1

MOs (essentially Is on C), the

rNiano9 searchbeing done over the next

8, Fhus he highest 12 virtual orbitals

%@~i-ot ttxaminad. Since the Ci calculations

@@IFerfarmr;d nly ORa portion of the

&M arhitz\lswe decided to estimate the

a%1 4%lencleurrrelntiunnergies of tiylobutane

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

PI.

2.2. Semi empir ical

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 ethyl-

enes 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 orig-

inal 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 amfigura-

tions 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

quotes his valueat 4 kczllmk


3/13

2.0 A ethylene-ethylene separation. The same

was done for the ST state at 2.8 A. This after-

CI 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,

3. Reaulls nnld discussion

3.1.

Gert eral met hodological onsiderat ior rs

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 molec-

ular systems with a large basis set, large CI

cnlculation can be trusted for a surface cal-

culation involving some changes in relative cor-

relstion energies along that surface [lo]. For

huge molecular systems, financial or technical

limitations impose smeller basis sets such ih: l t a

lorgc CI treatment may be difficult or presently

impossible and the resulting surfaces poor [IO].

Since both ab initio and semi-empirical cal-

culntions 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 semi-

empirical CI method over its ab initio counter-

part 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.u-

face calculations. The main reason for their

poor estimate of the relative ground state

energies of different molecules [12] hes been

established [13] and largely corrected at the

MCSCF level [14]. Likewise, we have demon-

strated that our reparameterized CNDQ CI

method yields adequate appearing ground and

excited state bond rupture surfaces [2f]. It

remains to be shown by comparative cal-

culations, however, whether or not a CNDO CI

method can generally mimic the excited state

surface features generated by an ab initio cal-

culation of similar basis size. Thus, part of what

.-Jill be explored here is a comparison of such

calculations and problems encountered.

3.2.

The et hyl ene-et hyl ene t ri plet dimeri zati otl

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,

13)

*Eoo hv = Euom, AH = 64 kcal;mole,

(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 Ego0 nd Ew are nearly iso-

energetic and thus the energy of reaction (2)


4/13

154

E. K assab et al. 1 theoreti cal anal ysis of he cycloaddi ti in of ethy lene

will approach the known activation energy for

thermal cis-trans isomerization of ethylene [lc].

Even though triplet sensitized photo-

dimerizations 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 obser-

vation WI

be conveniently rationalized [lg] by

examining the kinetic expression for the disap-

pearance of triplet olefin (3A) by competitive

unimolecular intersystem crossing

3A%1A,,

and bimoIecular dimerization

(5)

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 trip-

let-ground state intersystem crossing in

aromatics [17]. The near degeneracy of the trip-

Iet 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 com-

monly 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 reac-

tion pathway using geometry and spin transfer-

spin 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 CNDO-

UHF 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 trans-

3TMDR gave 1.44 8, for the -CHz-CHz dis-

tance, 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 reac-

tion (l), and 1.49 and 1.52 for the C-C dis-

tances 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 activa-

tion 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 cor-

f

See

also

ref. [k].


5/13

E. K assab et al . Theoretical nalysisof he cyclooddi ti on f ethy l ene

155

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 opti-

mized 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),

(8)

M = -16 kcal/mole.

(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 per-

formed geometry optimizations on ethylene in

which only two cis-hydrogens and the CC dis-

tance were allowed to vary and found an essen-

tialIy planar structure with the same CC dis-

tance 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 mini-

mum 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 coor-

dinate is the spin density. In both reactions

there is virtually no spin transfer at the tran-

sition state. Both the geometry and spin density

transfer criteria are consistent with the inter-

pretation that the transition state is more reac-

tant-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 pre-

condition 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 trans-

fer. 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 radical-

like 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 radical-

ethylene 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 cyclo-

pentene 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 reac-

tion 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


6/13

CARBQN-CARBON BOND DISTANCE, ANGSTROMS

Flfl. I. Rep~rnmcterizi zd [2f] CNDO4JHF calcul ations o f the additi on of tripl et 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 con stnnl al 1.08 A. Also sh own are the spin transfer from the radical or tripl et speci es 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.

lhus,

in a renl sense, ir is the variation in the

kilir values which differentiate the triplet sensi-

tized 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 experi-

mcntul 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 cth

yltme-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&e e1?3-

empirical 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 time S 51if-

face is in the 2.0-2.2 A regk~~

Qualitatively, the ab initio ay.xd e&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

OUT alcul2tions are diEerentia@d &om mrE+

calculations on the same system.


7/13

E. Kmab er al. f Th@orelfcal at&is

o lthz

yclnaddirion

of thylene

157

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

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 configura-

tions, 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

o f significence

singlet excited stare, S,

and dauhty excited state, S**.


8/13

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

I I

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 instabil-

ity will permit the formulation of diabatic

functions interconnecting the SOand S**.sur-

faces [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 ethyl-

ene-ethylene separations [26]. The configura-

tional behavior (table 2) of the So state is

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 MOs 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 MOs is obtained

which have a correspondance with those anti-

cipated from orbital symmetry rules. Un-

fortunately, the configurational composition of

the S* state at large ethylene-ethylene separa-

tions depends on whether one is working with

localized or delocalized MOs In the case of

? Note that this point is overlooked in thecorrelation

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

S**

SCF

CI

CI

1 IS 1.41 1.415 1.90 -155.562 -155.955

2 13 1.40 1.425 2.00 -155.640 -155.936

3 11 1.39 I.435 2.10 -155.700 -155.931

4 9 1.38 1.440 2.20 -155.747 -155.945

5 3.8 134 1.500 335 -156.875 -155.078

6 ethylene -17.952 -78.079

7 cyclobutaneb -155.961 -156.029

-155.745

-155.562 -155.767 -155.736 -155.866

-155.809 -155.639 -155.789

-155.757 -155.886

-155.838 -155.645 -155.792 -155.766 -155.893

-155.825

-155.644 - 155.787

-155.771 -155.889

-155.733

nonconvergent -155.723 -155.908

-77.477 -77.526 -77.672

-77.808 -77.852

SCF CI

T*

SCF CI

a)Experimental geometry.

) Planar geometry as found in 6-31G* calcuiation of Cremer [32]. Other geometries were taken from semi-empirical cal-

culations, 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_


9/13

E. Kassab eta . / Theoreti calanal ysi s of he cycloaddi ti on f eth yl ene

1.59

Table 2

Leading configurational terms in the ab initio calculations

Ri

Config.

coeKb =

(HOMO) (LUMO)

1.9

So

0.215 - 0.919

s**

=

0.900 + 0.200

2.0 So

=

0.379 - 0.860

.S**

=

0.846

-I-

0.359

2.1

So

=

0.662 -

0.665

Y*

=

0.539 + 0.639

2.2

SO

=

0.842 - 0.420

.S**

=

0.397

c

0.807

3.5

So

=

0.929 -

0.151

S**

=

-0.048 - 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.

localized orbitals (table 2) the double triplet

character of the proper (S2 = 0) four open shell

configuration is hidden within the poly-

determinatal 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 ethylene-

ethylene distances tends to become bitriplet in

character because only the central two deter-

minates contribute to the orbital-orbital

exchange term K. Likewise these central

determinates are triplet in character because

one has either both (Y or 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 polyconfigura-

tional 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

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% mono-

configurational). With regard to the semi-

empirical 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 arti-

fact of the CNDO method even under our

parameterization.

3.3.2.

Quant it ati ve aspect s-the correlat ion

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

ih

semi-empirical level [2a, 2d, 2f-

2i]. At the SCF level, the ground state thermo-

dynamics 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


10/13

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 4-

1 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 imptov-

brg 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 ener-

gies 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 cor-

rekition 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

lilhk I

~~~~IcIIII~B~nergy AimatCs fat ethylene and cyclobutane

__Ix_ -.-_..

h~


11/13

.G. Karsnb

et

al. / Theoreiicqt analpis

o the

cycloaddition

ofethykne

161

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 irrik ab

initib

basis

set

With regard to the CNDO-CI calculations it

must be pointed out that a similar size consis-

tency problem exists with regard to the CI This

can be demonstrated by examining fig. 3. Orig-

inally the overall thermodynamics of the

cyclobutane-two-ethylene reaction was cali-

brated at the 60x60 CI level by individual cal-

culations 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. ethyl-

ene) than larger (ethylene dimet) using the same

number of configurations. However, unlike the

ab initio calculations, a semi-empirical tech-

nique can be recalibrated under size consistent

conditions (i.e. cyclobutane and ethylene dimer

at large separations). Likewise, since the semi-

empirical-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 sepata-

tions the semi-empirical CI treatment must

include doubly excited four-open-shell

configurations. In principle, the inclusion of

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 pre-

liminary calculations at various points along ?he

reaction surface. Ir this mtmner, the semi-

empirical Cf treatment can be kept wlthin the

level 100 x 100. However, it must be pointed

out that the amount of semi-empirical car-

rel&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 1 llx

110 level, the latter containing all totally sym-

metric 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 semi-

empirical potential eaergy curve calibrated with

one set of configurations Will be out of calibr;l-

tion 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

VISUJ esemblance 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*k tates in the region of thy HOMO-

LUMQ crossing,

For the triplet diradical surface repara-

meter&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-empilicl

I :I calculations for the 2S+ 2S cyctoaddition


12/13

162

E.

Kamb et al. / Theoretical analysis of the cycloadditionfethylene

surfaces shows similar qualitative behavior.

e

demonstrate that the configurational behavior of

the S** state at large ethylene+zthylene dis-

taxes 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 tech-

niques 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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theoretical analysis of the cycloaddition of ethylene

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