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5612Table VIII. Experimental Value and Prediction of a(19F)Based on Equation 3 with Various Values of Q F p , QFc, and Q F ~ ~ a

Radical 23 24 SZCb HMb ExotbEq Eq2-Fluorobenzyl +12.8 +1 2 . 4 +11.1 + 5 . 9 8 .1 73-Fluorobenzyl - 5 .6 -6.0 -5.1 - 2 . 7 ( - )4 .874-Fluorobenzyl +13.1 +1 3 . 9 +1 2 . 4 +6.7 1 4 . 5 3

or The spin density obtained after spin annihilation by the INDOunrestricted SCF calculation was used as d or D in the equations.Taken from ref 33.

VI1 for their Q values). All of the m agree with ex-periments almost to the same extent. As Icli an dKreilick pointed out, l 3 the near proportionality be-tween psF and p ~and also P~CF n actual free radicalsmakes it almost impossible to determine reliable in-dividual Q values by fitting experimental a( 19F) againstspin densities. On the other hand, our results arebased on physical models that would retain the signifi-cance of individual Q values. Therefore it is not sur-prising that existing values which are already wellscattered did n ot agree with ou r values.

A few words of caution may be added to our re-sults. First, even thoug h our DZS set is anticipated togive a reliable overall picture, the individual Q valuescould be more sensitive to the choice of basis set. Also,our model of analysis using an artificially modifiedhalf-occupied T orbital is certainly a good way ofobtaining Q values, but it is not necessarily the onlyway of doing so. Different models may result in some-what different results. Also there is the lack of thequantitative agreement of experiment a(I9F) with thetheory for the CH 2F molecule. As the result anan arbitrary scaling factor of 2 had to be in t roduced .A better wave function may alter the interpretationsomewhat, but the qualitative conclusion would not beaffected.

Acknowledgment. The authors are grateful to Pro-fessor R . w. Kreilick for his constant encouragementand stimulating discussions, to Dr. R . M. Stevens forhis PO LY C A L program, and to Dr. D . M . Hayes forreading the manuscript. Partial financial support fromthe Alfred P. Sloan Foun dation is appreciated. Thenumerical calculations were carried out mainly at theComputing Center of the University of Rochester.

Calculation of Ground and Excited State Potential Surfaces ofConjugated Molecules.' I. Formulation and ParametrizationA. Warshel and M . Karplus*Contribution fr om the Department of Chemistry, Harvard University,Cambridge, Massachusetts 02138. Received January 31, 1972

Abstract: A form ulatio n is developed for the cons istent calculation of ground and excited state potential surfacesof conjugated molecules. The method is based on the formal separation of u and 7r electrons, the former being rep-resented by an em pirical potential function and the latter by a semiempirical model of the Pariser-Parr-Pople typecorrected for nearest-neighbor orbital overlap. A single parameter set is used to represent all of the molecularproperties considered; these include atomization energies, electronic excitation energies, ionization potentials,and the equilibrium geometries and vibrational frequencies of the ground and excited electronic states, and takeaccount of all bond length and bond angle variations. To permit rapid determination of the potential surfaces,the u potential function and SCF-MO-CI energy of the r electrons are expressed as analytic functions of themolecular coord inate s from which the first and second derivatives can be ob tained . Illustrative applications to 1,3-butadiene, 1,3,5-hexatriene,a,w-diphenyloctatetraene, nd 1,3-cyclohexadiene re given.

detailed interpretation of electronic transitions andA concomitant photochemical processes in con-jugated molecules requires a knowledge of the groundan d excited state potential surfaces. Th e deter min ationof such surfaces has long been a goal of theoreticalchem istry. Difficulties in a reliable a priori approach tothe problem for a system as simple as ethylene2 aresuch that calculations for more complicated moleculesare prohibitive at present. Consequently, a variety ofmethods that utilize experimental data have been in-troduc ed. Completely empirical treatm ents, in whichthe energy surface is expressed as a function of poten-tial parameters fitted to the available information

(1) Supported in part by Grant EY00062 from the National Instituteof Health.(2) U. aldor and I. Shavitt, J . Ch em. Ph ys . , 48, 191 (1968); R. J.Buenker, S . D. Peyerimhoff, and W . E. Kammer, ibid.,55, 814 (1971).

(equilibrium geometry, vibrational frequencies, etc.),have had considerable success in applications to mol-ecules for which a localized electron description isap pl i~ a bl e .~ he grea t advantage of th is type of ap-proach, which leaves open questions of reliabilitywhen extended from one class of molecules t o ano ther,is the ease and speed of the calculations; this had ma depossible applications to systems as large as certainnucleic acids and globular proteins. Fo r conjugatedmolecules, however, the importance of delocalizationintroduces difficulties into such an empirical trea tme nL5

(3) (a) See, for example, J. E. Williams, P. J . Stand, and P. v. R.Schleyer, Annu. Reu. Phys . Chem. , 19, 531 (1969); b) S.Lifson andA. Warshel, J . Ch em. Ph ys . , 49, 5116 (1968); A. Warshel and S . Lifson,ibid., 53, 8582 (1970).(4) M. Levitt and S.Lifson, J . Mol. B i d , 46, 269 (1969); M. Levitt,Nature (London), 224, 759 (1969).( 5 ) C. Tric, J . Ch em. Ph ys . , 5 1 , 4778 (1969).Journal of the American Chemical Society 1 94:16 1 August 9, 972

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5613to provide reliable results, the energy function must beexpressed in such a fo rm th at it an d its derivatives w ithrespect to coord inates and parameter changes can beevaluated efficiently. Fo r the u electrons, this goal isachieved by writing the potential as an analytical func-tion that consists of a sum of energy terms for the ap-propriately selected internal coord inates (bond lengths,bond angles, torsional angles, nonbonded distances,etc.). Th e a-electron potential is thus an empiricalfunction similar to those used previously for saturatedmolecules.3 How ever, som ewh at greater generality inthe potential is required here because of the largechanges in geometry that have to be encompassed in atreatment that is applicable to several electronic states,which can have significantly different equilibriumgeometries (e.g., the N and V states of ethylene). Fo rthe a electrons, the semiempirical configuration-in-teraction SC F-MO energy is developed as an analyt icfunction of the coor dinate s by means of a pertu rbatio ntreatm ent. Nearest-neighbor overlap, which is es-sential for obtaining a satisfactory description of theexcited states, l 2 is included by utilizing an orthogo-nalized (Lowdin) basis. l 3 Th e resulting expressions a rereduced to a convenient form by expanding them tosecond order as functions of the molecular coordinates.This permits a search for the minimum energy confor-ma tion and the determination of vibrational frequenciesto be made without prohibitive a mou nts of computationtime.The present treatment can be regarded as an exten-sion of the so-called consistent-force field (CFF)3bto conjugated molecules. In this appro ach the em-pirical potential is determined by choosing para met ersand functional forms such that the calculated valuesof molecular properties depending on the zeroth, first,and second derivatives of the Taylors expansion agreein a least-squares sense with the corresponding ex-perimental results. For finding the equilibrium geom-etry, a comb inatio n of steepest descent an d N ewton-Raphson procedures is used; the complete minimiza-tion with respect to all of the molecular coordinatesusually requires on the order of 40 iterations of theformer and four to six of the latter. The method hasbeen used prev ious ly for a lkane~3~nd, in a somewhatmore approximate form, for other molecules4~4 t h a tcan be described in terms of localized bo nds.In section I, the total energy of the molecule is ex-pressed a s a sum of u- and a-electron contributions andthe second-order a-electron SCF-MO-CI energy (in-cluding nearest-neighbor overlap) is formulated as ananalytical function of the corrdinates. Section I1 de-scribes the determination of the u and a parameters bya simultaneous fit to a wide range of experimental datafor ethylene, butadiene, benzene, and propylene. Th erequirement for consistency with ground and excitedstate results are shown to introduce significant con-straints on the parameters. In section I11 are given ap -plications of the method t o the ground -state prop ertiesof s-trans- and s-cis-butadiene, to the ground and firstexcited states of 1,3-cyclohexadiene and of 1,3,5-hexa-triene, and the ground state of a,w-diphenyloctatetraene.

Moreover, a completely empirical appro ach to theproblems un der consideration would appear t o requirea systematic method for introducing different param-eter sets for each molecular electronic state. Thissuggests that it would be better t o proceed by means ofone of the semiempirical procedures, of which there arema ny for s-elect ron systems. On e possibility is to usea method which includes all valence electrons ( e . g . ,extended Hiickel, IND O, PCILO , M IND O) .6 Al-though very promising, these treatments have not beendeveloped to th e state of refinement necessary to provideaccurate results for ground and excited state potentialsurfaces. Th e other possibility is to assume a separa-tion between u and ?r electrons and treat the u electronsvia empirical potential functions and the a electrons bya semiempirical appr oach . M any calculations of thistype have been performed and considerable successhas been achieved with appro priatel y chosen param -eters in the evaluation of ground -state properties e.g.,confo rma tions , dissociation energies).8 M ost of thiswork, which has used bond-order, bond-length relation-ships9 to simplify the treatm ent of the a-electron fram e-work, has been limited to an examination of the carbonskeleton with fixed bond angles and has ignored non-bonded interactions. Corresponding calculationshave been made for excitation energies, but these haveusually required large changes in the basic param -eters to obtain agreement with experiment, e.g., dif-ferent values of the resonance integral /3 in a Hiickelcalculation or the core parameter /3 in a Pariser-Parr-Pople treatment.gIn the present paper, we introduce a unified ap-proach to the ground and excited state potential sur-faces of conjugated molecules. Th e met hod is basedon a formal separation of the u and a electrons, withthe former represented by an empirical potential andthe latter by a semiempirical model of the Pariser-Parr-Pople type corrected for orbita l overlap. A singleparameter set is used to represent all of the propertiesconsidered : these include atomization energies, excita-tion energies, ionization potentials, and the equilibriumgeometries and vibrations of the ground and excitedstates, taking account of all bond-length and bond-angle variations. Th e use of the method for the ac-curate evaluation of Franck-Con don factors for elec-tronic transit ions is described in another paper.Applications to photochemical cis-trans isomerizationwill be given subse quently.The essential elements in implementing the presentmodel are the choice of the functional forms for thedifferent energy contributions an d the determination ofthe required parameters by comparison with experi-mental data . T o be able to include a sufficient rangeof parameter variation and a large enough body of data

(6) Fo r recent reviews, see R . Daudel and C. Sandorfy, Semiempiri-cal Wave Mechanical Calculations on Polyatomic Molecules, YaleUniversity Press, New Haven, Conn., 1971; G. Klopman and B.OLeary, Top . Curr. Chem. , 15, 415 (1970).(7) For recent examples of such calculations, see K. Machida, M.Nakatsuji, H. Kato, and T. Yonezawa, J . Ch em. Ph ys . , 53, 1305 (1970);J. M. M cIver, Jr., and A. Komorwicki, Ch em. Ph ys . Le t t . , 10,3 03 (1971).( 8 ) For a review, see M. J. S.Dewar, The Molecular Orbital Theoryof Organic Chemistry, McGraw-Hill, New York, N. Y., 969.(9) L. Salem, The Molecular Orbital Theory of Conjugated Sys-tems, W. A. Benjamin, New York, N. Y., 1966.(10) For some exceptions, see M. J. S. Dewar, and A. J. Harget,P r o c . R o y . Soc., Ser . A , 315, 443 (1970), and B. Honig and M. Kar-plus, Nature (London) , 229, 558 (1971).(11) A. Warshel and M. Karplus, to be published.(12) N. C. Baird, M o l . Ph ys . , 18, 39 (1970).(13) I. Fisher-Hjalmars, J . Ch em. Ph ys . , 42, 1962 (1965).(14) A. Warshel, M. Levitt, and S . Lifson, J . Mol. Spectrosc. , 33, 84(1970); S. Karplus and S . Lifson, Biopolymers, 10, 1973 (1971).

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5614I. Functional Form of Potential Surfaces

The potential surface of the Nth n-electronic stateVv(r) is assumed to have the form, as a function of theconfigurational coordinate rVAV(r)= V,(r) + V,O(r) + A V T N ( r ) (1)

where the a-bon d energy, V u @ ) ,s given by a n empiricalfunction, V,O(r) is the SCF-MO n-electron energy forthe closed-shell ground state, and AVT N ( r ) s the con-figuration interaction excitation energy for the Nthstate. In some cases e.g., highly twisted double bondsas in 90" ethylene), the groun d-state energy functionV,O(r) is corrected by a CI calculation that includes theessential double excitations.(a) electron Energy Expression. Th e x-electronenergy is calculated in th e Pariser-Parr-Pople (P PP )approximation corrected for nearest-neighbor overlap.The overlap correction is included because it is neces-sary for the proper torsional energy dependence ofdouble bonds, as is evident already from the classiccalculations of Parr and Crawford's and has been re-cently stressed by Baird;12 e.g., in the V and T statesof ethylene, a n-electron energy independent of angleis obtained in the zero-overlap PPP approximation.Since it is inefficient to include the overlap integralsexplicitly in the SCF -MO equations, we follow thep ro ced ur e of F i~h e r - H ja lm ar s ' ~nd use an atom ic basisconsisting of Lowdin orthogonalized orbitals (A). As-suming that the X basis satisfies the zero-differential-overlap conditon, we find the molecular orbitals

= Cu,(r)X, (2)P

with expansion coefficients G,,(r) and the a tomsdesignated by p, from the SCF-MO equations'F(r>v,(r) = d r ) v n ( r ) (3)

The matrix F (r ) has the PPP form in the X basis; that is

'FwY(r)= ' P r Y ( r ) l /J ' rv( r )AY,v (P vwhere P,,,(r) is the bond order

o c cPpY r)= 2 X ~ n p ~ r ) d r ) (5a)7,

and Q.(r) is the atomic changeQAr) = 2,- , ,(r)> (5bI

The s tandard notation of Pople16 is being followed ineq 2-5 except that the superscript X is used to designatequantities defined in terms of Lowdin orbitals and thevariable r indicates the dependence on the molecularcoordinates. T o obtain explicit expressions in term sof nonorthogonalized Slater orbitals, we make use ofthe relation

x ,, = s-'.'? s-112 = 1+ (I - S ) +3/g(I- S)2+ . . . ( 6 )where ? and x are vectors composed of the Lowdin and

(15) R. G. Parr and B. L. Crawford, J . Chem. Phys., 16, 526 (1948).(16) J. A. Pople, Trans. Faraday Soc . , 49, 1375 (1953); see also ref8 and 9.

Slater orbitals, respectively, and S is the Slater-orbitaloverlap matrix. Substitution of eq 6 into eq 4, keepingnearest-neighbor te rms thro ugh S2,,,+l, neglectingnon-nearest-neighbor overlap S,,,,*P, and making useof the Mulliken appro xim ation, yields the desiredexpressions. For the Coulom b integrals 'Y,~, the pro-cedure is straightforward and one obtains (suppressingthe configurational coordinate r)xYP , , = Y,, , + 1/2s2P,,+1(Y,,P YP,,+l) +

1/zs2,,,,,-l(Y,,,,- Y,,,-l)'Y,,,*:m = Y r , r * m m > 1)

x l , , p* l = YP,, ,*l - 1/2s2r,ril(YP,P Y,,,,il) (7)where the 7 without the superscript X correspond tointegrals over Slater orbitals. To develop expressionsfor the core integrals and W,, given by

'P, = 0 v = p 2 , p * 3, . . .P r v = '(PlHcore/v) v = P * 1 (8 )'Ww, = (P IHoo re lp )

with He,,, the standard core Hamiltonian,'e is some-wh at more complicated. We use relatio ns given byLowdin's in his classic papers on nonorthogonal orbi-tals and follow a procedure similar to, though not identi-cal with, that of Fisher-Hjalmars. l 3For A ~ , , , , * l , we neglect terms in S 2 and non-nearest-neighbor contributions to obtainI3,,,,*1 ' ( P I T + u,+ U,,*lIP * 1)

E P I T + u,+ U,*lIP * 1)'/2S*((PIT + U,IP) + (P * I l T + U,*lIP * 1)) ( 9 )

where U,, is the core potential for a neutral atom andS , = pip * 1). Introdu cing the potential Up+ or thepositively charged ca rbo n core(PIU,+lP * 1) = (PIU,lP * 1) -

( P P I P P 1) = PIU,lP * 1)l / z s * ( Y P P + Y,, ,P*l 10)where the approximate equality follows from the Mulli-ken approximation, we can write (eq 7 6 of ref 13)P,, ,+l = P + , , , i ls*[l/*(w,++ W+,il Y , , p i l l (11)

with the standard definitions of the Slater-orbital coreparametersPf,,,,*l = ( P I T + u+, U+,illP * 1) (12a)w+, (PIT + U+,IP) (12b)

To find an expression for W,, we proceed similarlyand write (neglecting terms in S 3 )' W , E (PIT+ u,, U,+i + U,-I + U Z I P )

Y,, ,,E (PIT+ u,,+ up+'+ Ur- i + UZIPL) i3/4(s2+s~-)(P/T+ u,l~>/@+(P + 1 / T + U ; y P + 1) +S2-(p IlT + U,-IlP 1))S+(PIT+ u,+ U,,+IlP + 1) -S-(P/T + u, U,-IlP 1) y r p (13)

(17) R. . Mulliken, J . Chim. Phys., 46, 497 (1949).(18) P. 0. owdin, J . Chem. PhJJS . ,18, 365 (1950); 21, 496 (1953).

Journal of the American Chemical Society / 94:16 1 August 9, 1972

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5615where nBP y , u rnd are the appropriate mutual polarabilities.9 Equations I 8 and 19 yield VTfl(r) as a con-tinuous function of r with con tinuo us derivatives, if theparameters themselves are represented by suitablefunctional form s (see below). Th e first two derivativesof the ground-state n-electron energy required for find-ing the minimum energy conformation and the vibra-tional frequencies will be obtained from these expres-sions. To evaluate the potential surfaceof the Nth excited state, the excitation energy AV;'(r)(eq 1) must be determined This is done by expressingthe configuration interaction energy of the excited stateas an explicit function of the coordinates r. We de-scribe here the formulation for a one-electron excita-tion to a singlet state; for a triplet state or for excita-tions involving more than one electron, the appro-priate modifications must be introd uced . Writing theexcited state *.v in the form

Excitation Energy.

= C C . d r ) (20)71

whereVn = l $ n 1 + n :

represents the singlet wave function corresponding tothe excitation from the SCF orbital nl t o / I ? and C.,.(r)is the vector of coefficients obtained from the secularequationslA(r)C,v(r) = A Vr.y(r)C.v(r) (21)

with( 4 n n = (1$n,-n21 HI $ n , - n ? ) W I$ )=

l 4 n m = (l$nl-n?lHl$ml-mJ =E , , E , ? (nln2 nlnr) 2 n 1 ~ ~ 1 ~ ~ ~ 1 )22a)

2(min?Im?nl) (mlnzl~ims) 22b)where(nmlkl) = s s a n ( )Qm(2)( r1.>Q il(l)aj,(2)d7,d7?=

C U , ~ P ~ / : P ~ r n p ~ ~ V h Y P23)P

(All coefficients are assumed real for simplicity.)To obta in AV;'(r> as a function of r, we proceed asfor the ground state n--electron energy, V, (r), an d ap -proximate AVr'v(r) in the neighborhood of the referencegeometry rs by use of the eigenvectors Css = C.v(rq).We haveA V, (r) E cs,iA(r)C.$ = x(Cs.vm)( lA(r)) +

m

2 CS. Sm(lA(r))mli = ~ ( C S ~ v m ) ? ~?(r) Em,(r)k > m m /C [ ( U s m l p U S m ? u ) 2 - 2 ~ ~ , , ~ ~ ~ m , u ~ ~ m ? / l ~ ~ m z v l Ih 3 P Y ( r )+PU \

where X y r r s given in eq 7 and U , represents the po-tential of the third l igand ( e . g . , another carbon or ahydrogen). Ma king use of eq 7, 10, and 12, we obt ain= WO, S + P t g p + l - S-P+,,,-1 +

s2 [3/4wfp ' /4W+r+l / l ( Y @ p Yr+ l , p + l )' / ? ( Y P P+ Y r , r + l l + SZ-[3/4W+r + l/IW+c-l

l / d Y P c Y P - l p - 1 ) l / d Y P , + YP , P - l ) I (14)where Wow s the ionization energy for an electron froma to m p , including the nearest-neighbor penetration terms

WflP W+, + /-IuP+i uc-i+ u z l ~ ) (15)For a conjugated n- system ma de up of identical atom swith the sam e nearest neighbors, eq 14 simplifies to

= w02,+ S 2 + [ - P + P P + l / S ++ w+z*l / ? Y P P + Y p + 1 > 1 + S2-[- W lis- +

W+2P 1/2(YP1.1 + Y P , P - l I (16)Ground-State Energy. Equations 7, 11, and 16 con-tain the expressions used for the basic parameters ofthe SC F n-electron theory (eq 4) in this paper. Oncethe bond orders P J r ) are determined by solving theSC F equations for a given geometry r, the ground-staten-electron energy VT0(r) f eq 1 is given by

V7r0(r) = C p P P ( ~ ) [ X W P ( r >1/4PPP(r1xYP/l(r)1P2 C P , m X P c Y ( r ) c '/zP2P (r>-

V > P v> PQP~r)QY(r)lxrPY(r>17)

To employ eq 17 for determining Vro(r) n the neighbor-hood of a given geometry r = Ti, we consider two ap-proximations. The f irs t consis ts of calculating thebond orders Pcy(rs) t rs and using their values for aneighboring configuration r ; i.e.V,O(r) = CPPP(rsPWc(r) l / J ' P P ( r s ) x ~ P P ~ ~ ~ lP

2C PcV(rs)XPaY(r)p> e

[ ' /2P2Jrs> - Qp(r s )Qdrs ) IX~Jr ) (18)v> c

Equation 18 yields the exact first derivative with respectto bo nd lengths or other parameters , a s can be demon-strated by use of the Hellmann-Feynman theorem orrelated methods.l9 This result is of particular im-portance for the direct use of semiempirical or a prior iSCF-LC AO-M O poten tial surfaces for classical tra-jectory calculations of reaction dynamics. To obtainaccurate results for the second derivative, which re-quires that the variation of the bond orders withand be included, we use the second-order ex-pressionV T o W= Ymo(r)l+ C C ( b P , Y / b X P u , ) r . ( h ~ P , Y ( r )

p> Y u> 7X P c v ~ r J ) ( h P u A r ~X/3Pv( rs) j+c c ~PP"/bXYu.)r,(XY,,(r) - xYc"(r4) x

('YUi(r) h Y u h q ) ) = V T 0 ( r > l+P> Y u> rn - p c . , A A P P ~ n / 3 u T ~ y P y , u , A X ~ P . A ' ~ u . ,19)r > u u > r I r > v r > r

c s m , P u s / ; l U S m 2 P ~ s / < l l X Y P v ( r ) (24)where u s m v = a,,(r,) (see eq 2). M aking use of thefac t tha t , to the same approximat ion

E m @ ) = vS,F(r)vSm (25 1with F(r) defined i n eq 4 and evaluated as are the cor-19) See, for example, R.Moccia, Theor. Chim. Ac t a , 8, 8 (1967).

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5616responding terms in eq 18, we can rewrite eq 24 in thef o r mAVT A i ( r )= C R W V X W , ( r ) C R Y Y Y Y Y Y ( r )

Y R P p Y X P , Y ~ r > R Y p v X Y r v ( r )26)

v > W v>rwhere the expressions for the coefficients R W Y , YYv, pYp,and RY , , of the various integrals are given in the Ap-pendix. The values of the coefficients, which depend onthe s ta te N , refer to the reference geometry rs, while themolecular integrals are determined for the geometry r .In section Ia,expressions w ere derived for the .rr-electron con trib u-tion to the ground and excited states by modifying thePariser-Parr-Pople method to include nearest-neighboroverlap. Th e resulting formulas depend on the valuesof the three integral parameters W(r), l(r), an dx y , , v ( r ) efined with respect to orthogonalized orbitals;the f inal equations for these p arameters are eq 16, 11,and 7, respectively. T o reduce these expressions toconvenient computational form, a number of additionalsimplifying assu mp tions were ma de. Penetration in-tegrals were neglected (except as their effect is subsumedin the empirical a-electron potential) and the distancedependences of several of the terms in the equationswere taken to be the same, as described below.we assumeth at /3*,,,*1 an d S, , , , , have the same distance anddihedral angle dependence and write the empirical for-mula as

(b) r-Electron Integral Parameters.

For the core resonance integral

PP,,*l = P o exP{ --P*.P(bp,r*l bo x(1 + kF@,,,*l b0)I x

[cos T,,p*l(l E,P,,,,*l cos 7,,,,,&1)1/[1- J , , , , , ,*lI (27)where 7 = +1+ z + r 2 + r 4) with r * the to rs ionaldihedral angles of the conjugated bond C,-C,*l.The first three factors in eq 27 represent the distancedependence of x31 for a planar system and thelast factor corrects for nonplanarity in terms of theeffective torsional angle 7 . Thus , h@,,,,*l depends onthe parameters Po, p p , bo l , okg , n d e,; b o 1was fixed atthe benzene value of 1.397 A.Alth ough the distance dependence of x/?,, ,I~ asexpressed in eq 27 is dom inated by the usual exponentialterm, i t was found necessary to introduce the morecomplicated function involving a two-term polynomialin the distance (or some similarly behaved function) tobe able to simultaneously represent the ground andexcited state potential surfaces. Correspondingly, al-though th e ordinary (cos 7 orsional angle dependencefor /3,,,+1 is a reasonable first approximation, itappeared that calculation of the torsional frequenciesrequired a correction term. This was introd uced bytaking account of the possibility of incomplete T -orbital following relative to the values of the variousX-C-C-Y torsio nal angles.*O Th e form used incor-porates a weak dependence on the bond order P,,,,*l;Le., the larger the bond o rder, the greater the tendencyof the T orbita ls to remain parallel, independent of theorientation of the adjacent bonds.

(20) For a discussion of orbital following, see D. M. Schrader andM . Karplus, J . Chem. Phjs., 40, 1593 (1964); D. M. Schrader, ibid., 463895 (1967); D. M Schrader and K . M o r o k u m a , Mol. Phjs. , 21, 1033(1971).

F o r W, as given in eq 16, the variation with distanceand torsional angle arises primarily from S2+and S 2 - .For these, we use the approximations2+ = S2,,,,+l = s20xs2- = s2,,,-1= s*ox (28)expi -2Pp(b,,,+1 bo)) cos2 7,,,+1

ex p( -2p6(b,,,,-1 - bo)} co s 2 T,,,-~which correspo nds to the lowest order contribution in-cluded in eq 27; here Sz0 s the value of the square ofthe overlap integral at a distance equal to bo1 . Ne-glecting other sources of distance dependence in w,,,we can write the simplified empirical form ulaW, , = Wozp+ P[exp( -2p,4br.,,+1 bo)} xcos27,,,+1 + exp ( -2pp(b,,,-1 - 0)1cos2~,,,,,-11 (29)The additional parameters required for W,, are, thus,Wozpand P, where the quantity P is introduced totake account of the factors in brackets in eq 16, as wellas Po.The formulas in eq 7 for the integrals Y,, ,. re ap-proximated correspondingly. The factors involvingappearing in Y,,, ,,n d Y , , , , , ~ ~ re assumed tohave the distance and torsional angle dependence givenin eq 28. As to the expression for the Coulo mb integraly p v tself, the formY,,= G exp( - ~ ~ b ~ ~ }e 2 / ( D+ bPJ,

D = e2 / G o (30)was chosen; here p., and G o re parameters and A and Iare th e valence-state electron affinity and ionizationpotential, respectively.8 Eq uat ion 30 is a com binationof a Nishimoto-Mataga type expression,* which givessatisfactory bond lengths in the ground and excitedstates, with an exponential, to provide the added flexi-bility required fo r other molecular properties. Avariety of other functional forms, including that ofO hno ,22 was tr ied and found to yield unsatisfactoryresults.Y,, ,, , I - A ) + GJexpf -2~p(b ,.+ l - bo)} X

G = I - A ) - Go,

Introd ucing eq 28 and 30 into eq 7, we obtainh

co sz 7,,,+1 + ex pf -2pp(b,,,,,-1 h }os2 7,,,,,-11G, ex p( -2pp(b,,,,*1 bo)} cosz 7,,, , ,+1

v + P , P * 1) (31)The new parameters are G (defined in terms of A , I ,and Go), Go, pr, and G,, which includes Sz0and thefactors in parentheses in eq 7; r,,, s the distance be-tween atom p and u .Substitution of eq 27, 29, and 31 into eq 18 (or eq17 an d 19) and eq 26 yields the com plete expression f orthe T-electron energies. It remains only to discuss thedetermination of the parameters appearing in theseequa tions. This is done in the following section; thevalues obtained for the parameters are given in Table I.

Y r , r i l = G xP ( --Plyb,,,,,*l} +-Y,,~ = G e x ~ [ - ~ ~ ~ , , , ~ le 2 / ( D+ r , J

+ b,,,*l) -h

(21) K . Nishimoto and N. Mataga, 2. h y s . Chem. Frankfurt a m(22) K. Ohno, Theor. Chim. Acta, 2, 219 (1964).Main), 12, 335 (1957).

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5617(c) +Electron Poten tial Function. Th e a-electro n

(32)where th e subscripts sat , conj, and sat-conj refer t o thesaturated, the conjugated, and the connect ion of thesaturated and conjugated parts of the molecule, re-spectively; i.e., Vuo(r)sat corresponds to carbon atomswith nom inally sp 3 hybridization, VuO(r)c,,nl to ca r -bon ato ms with nominally sp z hybridization, a ndVoo(r)sat--conl to the connection between the two. ForV,O(r)sat we use a slightly modified form of the alkanepotential function developed previously by Lifsonand Warshel. 3b It isVuo(r)sat= /?C[KO(ba- bo) + 2001 +

potential function Vuo((r)s writtenVuo(r)= r / ,o(r>3at V u o ( r ) c o n l + V Bo r )4a t - -CO nj

t

/ zCKe(Ba + /ZCF(qa yo +1 aCf(rt,)+ i12C~,(3)(1COS 34a) +

73 1C K . v ( e , ed(e, e, cos qh (33)where the subscripts i indicate the summa tions over allappropriate terms. The b f , O i , and + i epresent thebond lengths (CC and CH), bond angles (CCC andCCH), and torsional angles (XCCY), respectively;the q i are the 1-3 non bond ed distances, while the r f jare all higher order nonbo nded distances; and thepair O i and O i are two bond angles XCC and CCYof a CC bond. For each CC bond, K e ( 3 ) s chosento conform to the fact that only one torsional angleXCC Y (with X an d Y carbons) is included; for theterminal methyl group, all XCCH angles are counted.The Keet for XCCY con stants require inclusion of allpairs of atom s X a nd Y . The nonbonded function fwas chosen to b ef(rij) = Ae - Mr * l Brii-6 instead of thefunction 2 ~ [ ( r * / r , ~ ) ~3/2(r*/riJ6]+ e ie j / r t ,used in thealka ne potential. This modification allows us toemploy the same nonbonded function for the saturatedand unsa turate d parts (see below). It was observedthat the residual charge term e ie j / r i j can be omittedfrom the nonbonded potential without introducingsignificant errors for conjugated hydrocarbons.For VBo(r)conj ,e use the similar functionVoo(r)conj = C M b i )+ / l C [ k a ( a i QO) + 2DJ +i i

/ z CKe(e t- o ) z + /zCF qi 40) +1 i

C f ~ t j ) / ? C K , ( ) O S pi + 1/zCK,(2) OS 2 i +/ZCKX(Xi x 0 l 2 +C K e e / ( e i- e w i e o COS d f (34)

13 1 a

where the b i nd the ai epresent the CC and CH bondlengths, respectively, and the are out-of-planeangles defined for th e a toms A, B, and C at tached D as14

otherwise the nota tion is the same as i n eq 33. The CCbond potential is given by a Morse function of theformM ( b ) = Dh[exp(-2a(b b o } )-

2 e x p ( - a { b - bo})] (35)

Table I. Param eters for the n-Electron IntegralsIn tegra l Parameter Value

X P P -2.438 edVPi3 2 . 0 3 5 3-1kP 0.405 A 17 0 . 0 3bo 1 . 3 9 1 &

XW WO2, - 9 . 9 7 e VP 0.2 35 eV9.81 eV

Go 5 . 1 4 e VG, 0 . 6 9 e vPr 0 . 2 3 2 A-

XY I - A

instead of the quadratic expression in eq 33, becausethe distances for different degrees of conjugation andthe changes resulting on excitation are such that theharmon ic approxim ation is not valid; by contrast,the harmonic form can still be used for the CH bondsand for bond-angle bending. The angles r f include thefour torsional angles for each CC bond i n the conju-gated system; tha t is, for XIXzCCX1Xz, they are thedihedral angles XICCX1, XlCCX,, XzCC Xl, andXzC CX2 with 4%= 0,for the cis-planar geometry.The potential includes a onefold term (K,()) and atwofold term K,@))n these angles, i n contras t to thethreefold term that is dominant i n the saturated partof the molecule (see eq 33). All the other contrib ution sto the potential in eq 34 have the same form as in eq 33,though the constants are adjusted to take account ofthe differences between satur ated ( spa ) and con -jugated (sp*) carbon valences.For the term Vo r)5at-conl,he appropriate, somewhatsimplified, combination of the functional forms givenin eq 33 an d 34 was used.Vo(r)snt--conl= l / s C I K b ( b f- bo)* + 2 4 1 +Th e resulting expression is

a

/zCKO(O,- 0) + / z C F ( q z q d 2 + Cf(raj) +a 131/zCK,(3)(1- cos 344 ( 3 6 )Here the torsional angle r f is an C=C-C-X an gl e;the form of the function is such that the minimumoccurs with the double bond C=C eclipsing the CXbond , in ag reement wi th e~ pe r i m en t . ~ f X is a carbo na tom (Le. , except at a propylene end) only a singleI#J* angle is included to simplify the calculations; forthe propy lene en d, all the C=C-C-H torsiona l anglesare included with appro priate adjustmen t of the con-stants.The complete a-electron potential for each hydro-carbon molecule, relative to that of the dissociatedatoms, is obtained by introducing terms of the typegiven in eq 33, 34, and 36 for the bon ds an d their inter-actions. The m ethod a nd data used to obtain theparameters appearing in the potential functions aredescribed below with some discussion of the specialimp ortan ce of certain terms. A list of the completea-p ara me ter set is given in Table 11.11. Determination of Potential Param eters

The developm ent of section I has led to a formu lationfor the total energy surface V N ( r ) relative to the sep-

z

(23) S. Kondo, E.Hirota, and Y. orino, J . M ol . Specrrosc., 28,471,(1968).Warske l, Karplus I Potential Surfaces of Conjugated Molecules

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5618Table 11. Parameters for the u Potential Functionsa

c-c 8 6 . 0 1 10 1 . 4 9 0c p - c p 8 7 . 9 4 1 . 7 5 6 1 . 4 6 6cp-c 88.0 250 1 .450C-H 104 .0 286 1 .10 0C -H 104 .0 311 1 .09 0CP-H 103 .1 339 1 .080

c-c-cc 1 5 . 5 5 5 . 0 2 . 5 0CP-CD-CP 52.8 3 2 . 0 2 . 5 6C-C-H 25.3 4 2 . 9 2 . 2 0C-CP-H 24.0 2 9 . 5 2 . 1 8H-C-H 3 9 . 5 1 . 7 1 . 8H-CP-H 29.4 3 . 0 1 . 9

Out of plane / ? K xCP 10.2Tors iond /?K,+() /xK,+(*) I,I?K#($) Kea c-c-c-c 1 . 1 6 1 - 2 . 3x-cp-cp-x 2 . 3 0 . 6 6 - 6cp-c-pc-x 0 . 9H-C-C-H -9 .5

N o n b o n d e d A P Bc . . c 92431 3.60 747C . . . H 1 12 97 3 . 6 8 1 20H . . . H 1 64 2 3 .7 6 19

Bond angle* /?KO l ?F 90

C -C - H 1 8 . 3 5 1 . 7 2 . 2 0

a The units used are energies in kilocalories per mole, lengths iningstrom s, angles in radian s; the force constants are expressed cor-respondingly. Saturate d carbon atom s are designated by C andunsaturated c arbon a tom s by C p and , where differentiated frommethylene carbon a toms, carbons in methyl groups as C. *T h eparameters of C-Cp-Cp a nd C-Cp-H were set equal to those ofCp-Cp-Cp and CP-CP-H, respectively. Th e param eters of CP-C -Hwere set equal to the C-C-H parameters (ref 3). e For C C C a nadditional l inear term of t h e f o r m k e f ( @- 00 is used with KO =- 6.2 . d X a n d X may be either carbon or hydrogen. Thetorsional potential of the Cp -Cp bond is equally distr ibuted over allfour torsional angles. Th e torsional potential of the Cp-C isattr ibuted only to one torsional angle where X is the heavier a tomamong the three a to ms which are connected to the C s ide of thebond. Th e torsional potential of the C-CH, group is distr ibutedequally over all nine pairs X,-C-CH of the C-C bond.arated atoms) of a conjugated molecule in electronicstate N . Since the expression for V N ( r ) nvolves amixture of semiempirical and empirical concepts imple-mented in terms of rather complicated func tions tha tdepend o n many parameters, it is clear that the resultscan be regarded as meaningful only if they apply to asignificant number of properties for a variety of con-jugated hydrocarbon molecules. Also, the fittingprocedure itself requires considerable input informationand so can be carried out only if the necessary da ta ar eavailable.The method utilized for the molecular propertycalculation a nd pa ram eter determination is an extensionof the consistent-force field3b to c onjugated mole-cules. In this method, a least-squares procedure isemployed t o determine a set of parameters which yieldsatisfactory agreement between the calculated andexperimental values of molecular properties dependingon the zeroth, first, and second derivative terms in aTaylors expansion of the potential energy function.Since the details of the consistent-force-field procedurehave been given previou~ly,~be mention here only afew points that are of importance in its application toconjug ated systems. An essential element in the effec-tiveness of the procedure is the availability of analytic

expressions for the potential energy and the requiredderivatives. These can be obtained from the form ulasdeveloped in section I, as described below.The a-electron energy V,O(r), as given by eq 32,and its derivatives can be evaluated directly at eachpoint in coordinate space without excessive use ofcomputer time. However, the r-electron energy,V,O(r) + A V T N ( r ) ,was estimated by a simplified pro-cedure. For the steepest descent method employedin the initial stages of finding the minimum energyconfiguration, the first derivatives of the potentialenergy are required. Th e bond ord ers are calculatedat the beginning of each step and then eq 18 and 26 areused to determine the change in energy as a functionof the coordinates. Moreover, it is foun d tha t the newbon d o rders for each step can be calculated t o sufficientaccuracy by only a single iteration of eq 3. For themodified Newton-Rap hson met hod , the second de-rivatives of the potential energy with respect to thesystem coo rdina tes must be obt ain ed; tha t is, thechange i n coordinates, Ar, to go from the steepestdescent result tow ard the equilibrium value, is obtainedfrom the equation

(37)where V Vv(r) is the gradient an d Fs, the second deriva-tive matrix of the potential energy function for the Nthsta te of the molecule. Since F, is singular i n Car-tesian coordinates, the generalized inverse3 4 is used.Th e a-electron co ntribu tion to the matrix F.%, s obtainedby use of eq 17 for the ground state and includes thecontribution from eq 26 for the excited states. Oncethe equilibrium conformation is determined, the vi-brations are evaluated by finding the mass-weightednormal coordinates (2.v)I s linear combinations of theCartesian displacements from the secular equa tions

where 5,v = M- / F .vM-I with M a diagonal matrixcomposed of the atomi c masses.Use of the Newton-Raphson procedure is essentialfor obtaining an accurate equilibrium conformationsince the steepest descent method converges ratherslowly. I n practice, on the order of 40 steepest descentfollowed by four to six Newton-Rap hson iterations aresufficient to obtain convergence. Alarge number of experimental data were employed i nthe fitting procedure. They include the equilibriumcon form ation s and vibrational frequencies of the groun dstates of ethylene, butadiene, benzene, and propylene.Also used were the energies of formation of groun d-sta teethylene, benzene, and butadiene, as well as certainexcitation and ionization energies of ethylene andbenzene. In addition, the grou nd-st ate rotati on bar-rier arou nd the C=C bond i n ethylene and estimatesof certain excited state properties of ethylene and ben-zene were fitted. All these dat a are listed in Table 111,where the results obtained with the final parameter setar e given as well. It is seen th at satisfactory agreementhas been achieved for all of the properties.To determine the nonbonded interaction parametersin the exponential-six potential, an independent pro-cedure was used because the data included above are

(a) Empirical Data Used in Fitting Procedure.

(24) R. Fletcher, Comprrt. J . , 10, 392 (1968).

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5619

-5.0 I- 6 0 L I080 l o o 120 140 I60 1 8 0 200 2 2 0BOND LENGTH t i )

Figure 1 . Resonance integral as a function of bond length:(-) present calculation; (0-0) ref 27; (C--C) ref 28; 0-0)ref 26.

no t sufficiently sensitive to these interaction s. Th eproperties of n-hexane, n-octane, and some data forarom atic molecules were considered. Fo r all n-hexaneand n-octane crystals, the procedure described pre-viously was Fo r arom atic molecules, theresults of Williams2j were emplo yed; tha t is, his po-tentials for the interaction of two aromatic C H bondswere fitted by the present potentials over a series ofdistances and orientations. Reasonable fits were ob-tained, although Williams used origins for the inter-actions that were not centered on the atoms (in case ofhydrogen) while the present on es are centered on theatoms .The consistent fit achieved in Table I11 represents astrong requirement on the form of the potential func-tion and on the values of its parameters. No ne of theprevious semiempirical studies on a-electron systemshave tr ied to incorporate as great a variety of inde-pendent properties. The agreement between the calcu-lated and observed results gives some hope that theprediction of related properties in similar moleculeswill be of corresponding accuracy. It isof some interest to present the functions obtained forcertain of the a-electron integral parameters by thefitting procedure and to compare them with those usedby others. Since there is no correct functional formin such semiempirical theories, the variability foundin the different models is not surprising.In Figure 1, we plot the core resonance param eter @as a function of distance. Fo r comparison we includethe values of @ obtained by other workers with schemesthat neglect nearest-neighbor overlap and correlate amore restricted set of properties: they are the Pariserand Parr fit to excitation energies,2G the Longuet-Higgins and Salem fit to bond lengths and stretchingfrequencies,* and the Dewar, et al., f i t to atomizationenergies.A corresponding plot of the Coulom b integralsY @ , . n d X ~ r , r i l s presented in Figure 2 . Also in-

(b) Results for a-Electron Integral Parameters.

(25) D. J. Williams, J. Chem. P hy s . , 45, 3770 (1966); 47, 4680(26) R . Pariser and R . G . Parr, ibid., 21, 767 (1953).(27) H. C . Longuet-Higgins and L . Salem, Proc. Roy. Soc., Se r . A ,(28) M . J. S. Dewar and G .Klopman, J . A m e r . Chem. Soc., 89, 3089

(1967).251, 172 (1959).(1967).

2.00 0 I I I I J0 8 I 6 2 4 3 2 4 0 4 8

BOND LENGTH A )Figure 2. Coulomb in tegra l as a function of bond length: (-)Ay, present calculation; 0-0) ref 22; z-L) ref 21 ; 0-0) y,present calculation.

20.0 r

-200 01 0 0 120 140 160 180 200 220

BOND LENGTH A )Figure 3 . Bond energy contributions as a function of bond length(see eq 39 a n d 40): (-) total energy; 0-0) f rom O ( S 2 ) erm;0--0)rom M orse potential for u bond; (m-m) f rom 2xp; 0-0)f rom /. ( y r S r Y ~ , ~ + I ) .

cluded in the plot are the more standard expressionsof Mataga21a n d O h n o 2 e o r y r , . ; a similar comparisonis given in the recent paper of Beveridge and Hir~ze.?~It should be noted that tbe functional form of A ~ p , p + lis valid only to about 1.3 A, because for a smaller bondlength the simple exponential formula for the S 2 te rm(eq 28) overestimates the overlap contribution tox Y ~ , , , ~ Ieq 7 and 31). Inasmuch as the shortest CCbon d length of $erest in the present calculation isgreater than 1.2 A , this restr iction is not important;a modified expression, taking account of the true dis-tance dependence of S 2 , would be valid for smallerdistances as well.To summarize the various contributions to the dis-sociation energy of a conjugated bond, we plot inFigure 3 the results obtained for the case of unit bond

(29) D. L. Beveridge and J. Hinze, ibid., 93, 3107 (1971).

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5620Table 111. Experimental and Calculated Properties Used in the Optim ization of Energy Parametersa

A . Vibrational Frequencies (1/P = 50)Obsd Calcd Obsd CalcdA, 3026 2982 A, 3101 30801623 1654 3014 30631342 1321 3014 29871643 1680B1, 3102 3055 144 2 14561222 1177 1279 13031205 1228B2 3105 3070 890 877826 792 513 545B3u 2989 2993 3095 30941443 1452 3030 30623000 2988B1u 949 913 1599 16161385 1401B 2 g 943 965 1283 1303978 990A, 1023 1058 309 353967 999910 980Benzenec 680 679AI, 3062 3091992 1046 1014 1067A2g 1340 1389 909 932520 54 1B2, 995 102 5 170 176

Ethylene* s-trar.rs-Butadiened

703 61 5 Propylene&E2g 3047 3089 309 0 30871596 1614 3010 30621178 1149 2992 2988606 665 2954 29632933 2898El, 849 833 1652 16761474 1461Azu 675 668 1419 14441378 1434B I ~ 307 1 3093 1298 12941010 1068 1172 117196 3 999B2u 1310 1460 920 9341150 1158 428 4902" 97 3 997 2954 2962403 398 1443 14661045 1063El,, 3063 3087 99 1 10001473 1502 91 2 9511036 1046 578 578173 183

A

A

order, including the correction due to nearest neigh-bors i n the T system bu t neglecting Urey-Bradley an dnon bond ed interactions. The energy expression hasthe form (eq 17 and 34)E h o n d = v u o + vroE M ( b ) + (2'@ + 2'w, +

A/2(A3pp YP.,+l) 2W"ZP) = M ( b ) + 2 v +l /dYp, Y,, A+l) + @S2) (39)where the terms con tributing to O ( S 2 ) ,which arise fromW r , % and A ~ p f ~re

O(S2) = 2pte-2~~8(h--ha') l /sG,e-2fip J--h~') (40)It is evident from Figure 3 that the major p ortion of thedistance dependence comes from M ( b ) and '6, butthat the O ( S 2 ) erms are significant a s well.(c) Special Relationships between Potential Functionand Experimental Results. I n the fitting procedure,it was found th at certain of the experimental data (see

section IIa, above) could be duplicated only by includingspecific elements in the potential function. In theVosnt(r) nd Voconj(r)arts of VUo(r)he cross terms of theform Kse(Ot - eo)(&' - e, cos c ~ were found to benecessary to obtain the correct frequencies for thesymmetric B1, and an tisymm etric Bzu rocking mod esof ethylene. 3 0 , 3 These rocking frequencies are experi-mentally a t 826 and 1222 cm-', respectively, while thecorresponding calculated values in the absence of thecoupling term are 950 and 1100 cm-l; with the couplingterm, the values obtained are 792 and 1177 cm-I. Shi-manouchi attrib uted the large difference between thetwo frequencies to "the flexibility of the C=C bond"and proposed a special dynamical model for this.His model, which is different from the present one,represe nts a correc tion to the Urey-Bradley force field

(30) K . Machida, J . Chem. Phj s . , 44, 4186 (1966); W. L. Smith(31) T. Shimanouchi, ibid.,26, 594 (1957).and I . M . Mills, ibid., 40, 2095 (1964).

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5621Table I11 (Continued)

B. Othe r PropertiesObsd Calcd 1P Obsd Calcd 1 P

Conform ations Energy of formation'(a) Ethylene' (a) Ethylenec=c 1 . 3 3 5 1 . 3 3 1 0.005 Eo 5 6 2 . 0 5 6 0 . 4 2 . 0C-H 1 . 0 8 6 1 . 0 8 4 0 . 0 9C 4 - H 1 21 .3 1 20 .9 0 . 5 (b) BenzeneEo 1 3 6 4 . 4 1 3 6 5 . 8 2 . 0c=c 1 . 3 3 7 1 . 3 4 0 0,005 (c) s-trans-ButadieneC--C 1 .476 1 .479 0 .0 1 Eo 1 0 0 9 . 3 1 0 0 6 . 0 2 . 0c--c=c 1 2 2 . 9 1 2 2 . 2 1 .o

(b) s-trum-Butadiene0

c=c= C - C

'BzuFirst ionizationpotential' B z ~I&"'El"First ionizationpotential

(c) Benzeneh1 . 3 9 7 1 . 4 0 3 0.002 va(d) P ropylene

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5622e.g., a value for B?, of 1460 cm-I is obtained , while theE,, frequency remains in good agreement with experi-men t (1596 cm-' OS. he calcula ted value of 1614 cm-l).A difficulty in the present form of the met hod is thatthe calculated be nzene ring puckering B2, frequenc ytends to be too low ; it is near 600 cm-I while the ex-perimental valu e34 is 703 cm-'. By contr ast, othertorsional frequencies which have larger componentsof hydrogen motion (e.g., th e oth er B,, vibr atio ns)are calculated to be at somew hat higher frequencies thanthe observed values. This may result from the assump-tion that the angle between the pr orbitals on carbonsC and C' is related to a single 7 angle which is equallyde pe nd en t on =C-C=C-C= an d H-C=C'-Htorsio nal angles (see section Ib). The real situation isprobably tha t the T energy is affected more by thepuckering motion involving the =C-C=C-C=angles tha n the H-C=C-H angles. Th us, the dis-agreement may be due t o the f i t ting of all the tors ionalfrequencies with the same parameter, disregardingpossible differences between the orbital following.However, it does not seem worthwhile to introducesuch a refinement without more detailed considerationof U , T interaction.111. Applications

As tests of the present me thod, applications are giveni n this section to the ground states of s - t r a n s - and s-cis-butadiene, the ground and first excited states of 1,3-cyclohexadiene and hexatriene, and the ground stateof cy, w-diphenyloctatetraene. Th e 1,3-cyclohexadienemolecule is of interest because the presence of the ringleads to nonplanarity of the conjugated system. Thus,the molecule provides information important for theunderstanding of other sterically hindered conjugatedmolecules, such as the visual pigment 1 1 4 s retinal.Correspondingly, hexatriene serves as a simple modelof a large class of polyenes, of which a,w-diphenyl-octatetraene is another important example.Butadiene. Alth ough a variety of dat a for butadienewas used in determining the parameters, the differencein energy between s-cis and s-trans was not included.We have calculated the minimum energy conformationsfor the two geometries (see Table IV). We see thatthe bond lengths and angles are similar, but that thereis a significant increase in the C1C2C3angle of s-cisrelative to s-tra ns tha t relieves the steric repulsion inthe former . In contras t to Dewar and Harget , l0we find that the s-cis geometry does correspondto a shallow energy m inimum ; that is, the secondderivative matrix F is positive definite for thisgeometry. As to th e energy differences, the s-transconformation potential energy is 0.6 kcal lowertha n tha t of s-cis. When the zero-point vibrationalenergy is included, the energy difference becomes1.O kcal, to be compared with the experimental estimateof 2.3 kca1.3G Since the latter figure involves a varietyof assumptions, it would be useful to have an improveddeterm ination . Th e potential energy barrier to go froms-trans to s-cis is calculated to be 10 kcal relative tos-trans, as compared with the estimate of 5 kca1.36Table IV also includes the vibrational frequencies forSer . A , 191, 39 (1947); 193, 456 (1948).J . Ch em. Ph ys . , 14, 67 (1946).

(35) C. A . Coufson and H. C. Longuet-Higgins, Proc. R o y . Soc.,(36) J. G. Aston, G. Szasz, H. W. Wooley, and F. G. Brickwedde,

Table IV. s-cis-and s-trrrns-1.3-Butadienes-trans s-cis s-trans

Calculated Geometry&Ci-C, 1.34 15 1 342 9 Ct-Ca-HI 1 2 0 . 3C r C 3 1 , 4 7 9 0 1 . 4 7 4 8 C r C i - H i 1 2 1 . 7CI-H, 1 ,0856 1 0860 Ci-CrCo 1 2 2 . 2CI-Hi 1.0851 1.0836 CI-CI-H, 119 .9Cr-Hr 1.08 48 1.08 65 Cr-Ca-H3 1 1 7 . 9Calculated Vibrational Frequenciesb

3063 3063 9802987 2988 6801680 1676 B" 10671456 1456 9321303 1305 5411228 1083 176

A, 3080 A i 3085 B, 999

877 884545 354A 3094 BL 30833062 20632988 29891616 16401401 14421303 1278990 1057353 633

s-cis1 1 9 . 71 2 2 . 81 2 5 . 31 1 9 . 31 1 5 . 4

A1 1035992523

1039690207

Br 1093

a Distances are given in dngstroms; bond angles i n degrees.Vibrational frequencies are given in reciprocal centimeters.

the two geometries. Mo st of the corresponding fre-quencies are very similar, as expected. How ever,important differences do occur for some frequenciesthat have large contributions from the tors ion about thesingle bond or from the bending of angles involving thesingle bond; e.g., the lowest frequency of each sym-met r y ty p e an d th e 1 2 1 0 - ~n i - ~, vibration i n s-trans,which corresp onds to 1081 cm-I (A1) i n s-cis. Thes-trans experimental results are given i n Table 111since they were used in the param eter fit; no s-cisdata are available for com parison.One point to note i n the experimental comparisonis that somewhat more exact agreement for vibrationalfrequencies with specifically fitted force fields can beobtained,37 o that the present approach is not the oneto use if one is concerned simply with the vibrationalanalysis of molecules of know n structure . It is forthe calculation of a wider class of properties, as wellas for simultaneous determination of an unknown ge-ometry and vibrational frequencies, that our more gen-eral treatme nt is most suitable.1,3-Cyclohexadiene. Th e ground-s tate equilibriumwas calculated for 1,3-cyclohexadiene and the resultsare given in Table V. Two calculated geometries areincluded because the difference i n energy between themis found to be very small; that is, the calculated po-tential function has a flat minimum for a distortedhalf-chair conformation and changes i n the c1c;,c6c1dihedral angle of -10" yield a difference of only 0.1kcal/mol. Figure 4 illustrates this result i n a two-dimensional contour diagram, which gives the energy(kilocalories) as a function of the torsional anglesC1C2C,C4(42-3)n d c4 c>C g c1 +j-G). t can be seenthat there is a large flat basis for +2.-;j in the region8-16' and 4i-6n the region 25-42", The dom inant

(37) E . R . Lippincott, C. E. White, and 1. P. Sibilia, J. Anrer . Chem.Soc., 80, 2926 (1958); E . R. Lippincotr and T. E. Keiiney, ibid., 84,3641(1962); E. M . Popov and G. A. Kogan, Opr. Spectrosc., 17, 362 (1964);R. M . Gavin, Jr., and S. A . Rice, J Chem. Phj .s . , 5 5 , 2675 (1971).J o u r n a l of the Amer ican Ch emica l Socie ty I 94:16 1 August 9, 1972

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5623Table V. Ground-Sta te Proper t ies of 1 3-Cyclohexadienea

GeometryExperimental Calculated*c d e min (1) min (2)c1=c* 1 . 3 3 9 1 . 3 5 0 1 . 3 4 8 1 . 3 4 7 1 . 3 4 6c2-c3 1 . 4 6 8 1 . 4 6 8 1 . 4 6 5 1 . 4 7 1 1 . 4 7 5c4-c5 1 . 4 9 4 1 . 5 2 3 1 . 4 1 8 1 . 4 9 0 1 . 4 9 2c5-c 6 1 . 5 1 0 1 . 5 3 4 1 . 5 3 8 1 . 5 4 0 1 . 5 4 8Ci-C2--Ca 12 1.6 12 0. 1 12 0.3 12 0. 0 12 0.5C B - C ~ - C ~ 1 1 8 . 2 1 2 0 . 1 1 2 0 . 2 1 2 1 . 4 1 2 1 . 8C4-C5--C6 111 .5 110 .7 110 .9 11 2. 1 11 3.4Ci-C*-Ca-C( 1 7 .0 18 .3 18.0 12 11cI-c5-c6-c1 46 40 34

Vibrational FrequenciesCalcd Exptlf,g Calc d Exptlf.8B 3090A 3089B 3087A 3086B 2982A 2964B 2947A 2937B 1680A 1635A 1460B 1442A 1438B 1430B 1408A 1318B 1274A 1259

30573043305730433009*293828752859160415751460*142814391408137213261316*1239

A 1208B 1175A 1117B 1028A 1005B lo00A 990A 961B 951A 884B 784A 718B 652A 61 1B 529A 49 1B 312A 160

1177*116511501058101899494793185074875566356147850829820 1

a Th e units are bond lengths in Angstroms, bon d angles in degrees,an d vibra tional frequencies in reciprocal centimeters. See discus-sion in text . cRefe rence 38. dRefe rence 39. cRefe rence 40.f Reference 44. 0 The frequencies with an asterisk differ in assign-ment f rom tha t in footnote 1;. see text.

interactions leading to this form for the potential func-tion are the competit ion between the tendency of the?r system to be planar and the stabilization of a non-planar geometry by the CC C ring angles and n onbonde dhydrogen repulsions. Th e com paris on with electron-diffraction experiments 38- 40 shows satisfactory agree-ment for all bond lengths and bond angles . Thereare a n um ber of a ppa ren t differences but it is difficult todetermine their significance because of the large dis-agreements among the different experimental results.A s to the to rs ional angle CIC2C aC4,he electron-diffrac-tion results are interpreted to give a value of 17-18"and a partial microwave s tructure determ ination4I yields17", all of which contrast with the calculated value of11-12", The re is also a value of 14 f r o m an X- r aystructure for the 1,3-cyclohexadiene ring in the anti-b i o t i c g l i o t o ~ i n . ~ ~s has been pointed out in theanalyses of Traetteberg40 an d of the valueof the torsional angle obtained by interpreting theelectron diffraction or microwave data depends on theassumption of planarity for the two ethylenic groups,nonplanarity leading to a lower tors ional angle. Ourcalculations suggest that deviations from planarity of

(38) G . Dallinga and L. H. Toneman, J . Mol . Struct . , 1 , 11 (1967-(39) H. Oberhammer and S. H. Bauer, J . Amer . Ch em. Soc., 91, 10(40) M. Traetteberg, Acta Ch em. Scartd., 22, 2305 (1968).(41) S. S. Butcher, J . Ch em. P h ~ s . ,2, 1830 (1965).(42) A. F. Beecham, J. Fridrichsons, and A. McL. Mathieson,

1968).(1969).

Tetrahedron Lett . , 27, 3131 (1966).

$5.6

Figure 4. Calculated contour m ap for 1.3-cyclohexadiene showingthe energy (kilocalories) as a function of the torsional anglesC1C2C3C4 and C4C,CbCl $,-$): all other degrees of freedomhave their minimized v alue for eac h choice of and p --6.

- 5 can occur. Also because of the rather flat formof potential discussed above, the calculated root-mean-square dihedral angle (at room temperature) is 14".Thus, it is clear that both the analysis of the experi-ments and the comparison with the calculations arecomplicated by the possibility of large amplit ude motio ninvolving several degrees of freedom.Th e calculated vibrational frequencies for 1,3-cyclo-hexadiene are also given in Table V and compared withthe measurements of DiLauro, Neto , and Califan o.44The agreement between the calculated and the observedvibrational frequencies is generally good, though notquite as precise as that obtained by DiLa uro, e t al.,who used a valence-force field to fit the results for 1,3-cyclohexadiene. There is also some question con-cerning assignments. Ou r assignment of the experi-mental vibrational frequencies is similar to tha t ofDiLauro , et a l . ; the differences, which are indicated i nTable V , correspond to cases in which combinationbands of DiLa uro, et al., are interpreted as fundame ntalsand/or a different symmetry choice is made; the systemis such that both A an d B vibrationals can be infraredand Raman active and that combination levels can bequite strong. To make a definitive interp retatio n of thevibrations, isotopically sub stituted species are needed.It should be noted that the lower frequency vibra-tions (490, 300, 180 cm-') in 1,3-cyclohe xadiene may allbe rather inaccurate in the harmonic approximat ion .Of these, the lowest (160 cm-I) corresponds to theC4C aC6 Ci ors ion, the next (300 cm-') to r ing foldingof C1C2C,C4 elative t o C 4C; C6C ~, nd t he highest (490cm-l) to C,C2C3CI orsion. All of these have doubleminimum potentials for which the barrier at the"planar" geometry is rathe r l o w ; e . g . , it is I kcal for the160-cm-' torsion (see Figure 4). A more correcttreatment would thus introduce a quartic oscil latoror similar appr oxim ation for these vibrations. Thereis the added complication th at there is also a very s tronganharmonic coupling between torsional and bendingmodes. A similar effect of strong anharmonicityapp ear s in other sterically hindered conjugated mole-cules e.g., 11-cis retinal) and may play a n im portan trole in determining their vibronic structure.Ground State of Hexatriene and a,o-Diphenylocta-tetraene. The calculated ground-state structures of

(43) For an example of the treatment of Iargc amplitude motion i nthe interpretation of electron diffraction results, scc A. Yokozeki,K. Kuchitsu, and Y . Morino, Bull. Client. SOC. rip. , 43, 2017 (1970).(44) C . DiLauro, N. Neto, and S . Califano, J . M o l . St r r i c f . , 3, 219(1969).Warshel, Karplus 1 Potential Surfaces of Conjugated Molecules

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56241,3,5-all- trans-hexatriene an d of CY,w-diphen yl-all- trans-octatraene are given in Table VI with the resultsTable VI. Ground-State Structure ofHexat r iene a nd Diphenyloc ta te t raenea

-1,3,5-Hexatriene--Exptlb CalcdC1-Cz 1 337 1.340cZ-c3 1.458 1 .477c3-c4 1.368 1.350C1-C*-C3 121.7 122.1c2-c3-c4 124.4 122.0

Th e calculated ring torsion angle (2.5') is somewhatsmaller than the experimental value (5 ); whether thisis a real difference or a crystal effect is not clear sincethe potential function corresponds to a rather shallowminimum.The calculated vibrational frequencies of 1,3,5-hexatrienes ar e compa red with the experimen tal values37in Table VII. Th e agreement is similar to that forbutadiene, which was used in the fitting procedure.This "conservation" of agreement implies that cor-respondingly accurate results are expected to be ob-tained for other polyenes.Table VII. Vibrational Frequencies of Hexatriene"

ExptP Calcd Exptlb CalcdA, 3085 3091 B" 3091 30973039 3078 3C40 308 13039 3062 3012 30622989 2987 3012 29871623 1689 1623 16571573 1606 1429 1448ExptP Calcd Expt lC Calcd 1394 1420 1294 1311

1,8-Diphenyl-l,3,5,7-octatetraenec

1.3911.3871 4041,3981 3921.4051.4681.3501.4381 3591.441

1 4021.4031.4151.4031.4001.4151.4821 . 3 5 31.4681,3551.467

118.2120.9121.1119.7122.7119.2126.8123.9124.8124.54 . 9

18.6120.3120 .6119.8122.6118.6125 .1121 o122.4121.82 . 5

a The uni t s a re bond lengths in Lngstroms and bond angles indegrees. See ref 45. See ref 46.

of experimental measurements. 4 5 1 4 6 T he CH bondlengths and the CCH bond angles are not in-cluded because unique experimental values are notavailable. Fro m the table, it is clear tha t the generalagreement is satisfactory. Fo r hexatriene the order-ing of the bond lengths is given correctly by the calcu-lations, although the two "double" bonds are closer toeach other in length than the experimental values andthe "single" bon d is somewhat t oo long. Also, themeasured value of C2 C3 C4 s significantly larger th anthe calculated result. Fo r diphenyloctatetraene, boththe ring and polyene chain bonds are well reproducedby the calculation. Th e calculated bond angles showgenerally reasonable agreement with experiment exceptin the middle of the chain Le., cSC9cl0, CsC&o'),where the calculated values are somewhat smaller thanthe observed angles.

The most important point about including variationof bond angles in the calculation is the possibility ofreducing steric repulsions by bond-angle distortion.Previous calculations of polyenes properties haveassumed fixed bond angles so that steric effects mani-fested themselves only by changes in torsional angles.By minimization of the energy in the complete con-form ation al space, the additio nal effect of bond-anglebending is intro duce d. An example of this is evidentin the CsC7Cs angle, w hich is very large (exp tl 126.8',calcd 125.1') due t o the C3,CS hydrogen repulsion.(45) M . Tratteberg, Acta Ch em. Scand. , 22, 628 (1968).(46) W. Drenth and E. H. Weibenga, A c t a C r j , s t a l fogr . , , 755 (1955).

128012381187897444347990928897758C395c

13531307121694 1462393I03599086 I603231

A

1255113094 154OC

101194 1899658475c

12901185947587179

10899509226422471ns_

a All frequencies in reciprocal centimeters. See ref 37. Thereis some question as to whether these frequencies are correctlyassigned.

Excited States of Hexatriene and Cyclohexadiene.The details of the application of the present method tothe excited states of conjugated molecules will be de-scribed in a subsequent paper together with analyses ofthe vibronic structure of electronic absorption bands.To provide an illustrative example of excited-statecalculations, we present in Table VI11 the calculatedconformation and vibration frequencies of 1,3,5-hexa-triene and 1,3-cyclohexadiene in their first-allowedexcited electronic states. Com pare d with Table VI,we see that the most striking change in conformationon excitation of 1,3,5-hexatriene is the increase i n lengthof the double bonds (ClC2 and C,C,) and the shorteningof the single bond (C2 C3 ); there is little change in thebon d angles. The re is also a significant reduction inthe double-bond stretching frequencies e.g., the A,grou nd-st ate frequencies 1689 and 1601 cm-l). Theseresults agree, as will be shown," with the Franck -Condon factors and vibrational bands observed in theelectronic spectrum. 47For 1,3-cyclohexatriene, similar changes in structureand vibrational frequencies are calculated. As ex-pected, the C2Casingle bond is significantly shortenedand the torsional angle is reduced. There are no quan -titative experimental results for comparison becausesuch a distorted, s-cis type of molecule has a diffusevibronic struc ture in the excited state .47

(47) H. chuler, E.Lutz, a n d G . A rnold , Spectrochim.Acta, 17, 1043(1961).

Journal of the American Chemical Societ y / 94:16 / August 9 , 1972

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5625Table VIII. Calculated Geometry and Vibrational Frequencies of Hexatriene and Cyclohexadiene in First Excited Electronic State4

Conform ation Vibrational frequencies1,3,5-Hexatriene A, 10001c2 1.389 A, 3090 B, 3093 B, 953

c 3 1 . 4 0 9 3078 3082 850 922c3-c4 1 . 4 3 6 3062 3062 839 905C1-Cn-Ca 1 22 .4 2984 2984 553 597Cz-C&4 121.3 1596 1590 274 2051518 1466 1111396 13191345 1262

1292 12421203 933944 566453 1733861 . 4 4 21.3881 . 4 8 01 540

1 1 9 . 81 2 2 . 31 1 1 . 8

530

1,3-CyclohexadieneB 3090A 3088B 3084A 3083B 2984A 2964B 2948A 2939A 1600B 1543A 1457B 1444

A 1437B 1427B 1399A 1310B 1278A 1250A 1200B 1170A 1125A 1021B 985A 967

A 961B 947A 878B 828B 758A 690B 638A 592B 518A 437B 198A 160

The units are bond lengths in angstroms, bond angles in degrees, and vibrational frequencies in reciprocal centimeters.