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kinetic, potential, and total energies takes the particularlysimple form given by eqn. (1) 5)

where E is the total energy,T the kinetic energy, and V t h epotential energy of the system. For two differe nt stationa rys ta tes

From 2), we see tha t E = - AT, and a process tha t lowersthe kinetic energy actually causes an increase in the total en-ergy of th e system. O nly an increase in the m agnitud e (low-ering) of the potential energy reduces the total energy andgives rise to stabilization. Therefo re, in a reaction, if on e ;s toattr ibute enhanced stahil i ty of transi t ion states, intermedi-ates, products, or rea ctants t o delocalization, this favorableeffect must he due to a more favorable potential energy si t-uation.

Consider, a s a case in point, covalent-bond form ation fromtwo atom s or radicals

R. + R.' - RSince t he process is favorable,AE is negative. Consequently,

the change in the k iset ic energy,AT, for the process mu st beoositive and unfavorable. In t he case of two hvdroeen atom s.for rxnmplv, hond formntion iiartuallyn Iwalva tion process.

The two electrons are more lhralized in the hvdroren moleculethan they are in the separated atoms.~ d tn h does eachelectron in teract with the other nucleus, it also interacts m orestrongly with its own nucleus. In a ma them atical sense thisis reflected in an increase in the orhital expon ent of th e hy-drogen 1s orbitals during hond formation. This increasedexponent shrinks th e size of t he orbitals, increases the kineticenergy of th e electrons h ut also increases th e magnitude of thepoten tial energy; enhanced stahility results.

Bond formation in the hydrogen molecule localizes theelectrons an d leads to an increase in th e kinetic energy of thesystem; yet the potential energy becomes more negative, andthe latte r effect imparts stahility to the molecule6).

Heitler a nd London 7 ) n their classic treatm ent of thehydrogrn m olrcule did nor nlter th e whirnl exponent of the

s orhtnle, and whilt: their calrulnted enerrv for bond for-..m a t i m is quire y

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Figure 2. Top view of d i a p r n In Figure 1

The water molecule represents another interesting example.

The s tructure is usually written a s

with th e nonbonded electrons assumed to occupy slightlymodified sp 3 hyhrids. These, very approximately, look asfollows

c+/

This picture leads to the expectation of a bimodal electronicdistribution; yet a picture of the t otal electronic density in theplane of th e nonbonded pairs h s been published and ac-tually appears as shown in Figure 1 or, alternatively, as inFigure 2 13). Ther e is no bimodal distribution.

This result also has interesting ramifications for moleculeslike methanol which exist in th e geometry 14)

..oI

H

If one uses the localized picture for the nonbonding pairs,this s tructure corresponds to a completely staggered confor-mation with a carbon-hydrogen hond in the region betweenthe lone-pairs on the oxygen atom, but t his satisfying picture

is lost if one uses a more accurate description such as the onejust presented for water.The pictures we present in the introductory organic

chemistry courses m oth simple and appealing and generallylead t o correct predictions ahou t th e behavior of organic sys-tems. Nevertheless, they do not hold up und er rigorous scru-tiny.

Th e idea of localized lone pairs in nonbonded hybridizedorbital s is so well ingrained and works so successfully tha t i tseems worthwhile to examine more closely some of the pre-dictions of this approach, for even here the conclusions are notobvious. Consider, for example, which type of nonhondedhybrid-sp, sp2 , sp3-gives rise to th e largest dipole mo-ment.

A hybridized nonbonding orbital h can be written as2

h = 1 + h2)-' (S + Ap)where A takes on t he values 3lf2, 2112, and or th e commonsp sp2, and s p hybrids. Th e dipole moment of an electronin such a nonbonded hybrid is

p = jhedhdu

where e is t he electronic charge and d its distance from thenuclear center.

Subsituting for and expanding leads to1

p = J(s + Ap)ed(s + Apldu1 t A

2This derivation was presented by Dr. Ruben Pauncz at theQuantum Chemistry Institute given at the University of Florida,Gainesville, in January 1969.

..The f i t ntegral inside the brackets in eqn. 3) is the dipole

moment of an electron in ap ur es orbital while the last is th atof a n electron in a purep . We know these to be zero since thereis no dipole moment associated with pure s and p orbitals.Therefore. the d iw le moment associated with the hvbrid mustcome from the cioss-term. Furthermore, the maximum valued e ~ e n d s ot on the internal. which is const ant for all hvbrids..~ ,.hut on the value of A. We m u t simply maximize the functionX I 1 + h ) tn find whirh tvuc of hvhrid has assoriated with it.the maximum dipole moment.

We take th e derivative and equa te it to zero

The electron has a maximum dipole moment when oc-cupying a hybridized nonbondingorhital having A equal to ,and these correspond to the two sp hyhrids. I t may be of in-terest togive the values of theq uant ity U 1+ A2) for the morecommon values of A, and we do that in Tahle 2.

Th e last value inTahle 2 is obtained

Table 2. TheValuesaf h l 1 + A 2 for either by usings sp, sp , sp: and p orbitals L'HBpital's rule or

by rewriting the( '

quantity as I/(X +1/X and letting A

: approach infinity.3 0.433 Starting with pure s

orbital , mixing in pcharacter bringsabout a rapid in-

crease in the dipole moment. The average position of theelectron moves away from th e nucleus. The maximum occurswhen equal a mounts of s and p have been mixed. Continuedincrease i n p character gradually reduces the dipole moment

back t o zero. I t may come as a surprise that t he maximummoment is associated with the s p hybrid rather t han t he sp3.This result and some of those stemmine from delocalizationarguments indicate that even our most basic assumptionsshould continuallv he reexamined. For example, do the sixr-el ectr ons in beizene really acruunt for its s ~ n h i l i t ~ , r arrthe 36 o-electrons actuallv responsible I f ? ) ?Even tndnv, thisvery fundamental cannot he answered with cer-tainty.

Conclusion

Th e ideas iust oresented are neither verv difficult nor verymathematical, A d th eir introduction -into the organicchemistry course would no t tak e much time. They may givethe student a better idea of some fundamental concepts.

All electrons a re delocalized, and such delocalization, if itis truly t o account for the difference in energy between s ta-tionarv stat es. must re sult in a lowerinn of the po tential en -ergy. The kinetic energy must increase with increased stahilityof the svstem. Alternatively, certain processes, considered tohe del&alizations, may actually bd localizations, e.g. , theformation of the covalent hond in the hydrogen molecule.Even then, the kinetic energy must increase, and greaterstability must come from a lowering of the potential ener-gY

Acknowledgment

This work was done a t th e Universitv of California.Berkeley.

The author is verv much indebted t o the Facultv of theChemistry ~ e ~ a r t m e n t t the University of ~a iif orn ia,

Volume 54, Number 8. August 1977 1 481

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Berkeley and especially to Professor A. Streitwieser Jr. for I 2:; ~:~&~~, ~; i l l ,Nwthe gracious hospitality extended to him during a Sabbatical ~ o r k . 963, Chap ter 3.year spent there as Visiting Professor. ( 7) ~ e i t l e r , ., a n d h n d o n ,F ~h ys ik , 4.4% (19271.

(81 W sng, S. C., Ph ys R a u . 31.579 (1928).(9) Rosen N., Phyr. Re .. 38,2099 (19311.

Literature Cited (10) Eile.8, J. E.. and Liberlr *.A J Arne? C h e m Soc. 97,4183 (19751.(11) Liberles.A. Greenherg A..and Eilers, J.E., J CHEM. EDUC.,60,676 (19731.(1) Liberles, A , lnt rd ud ion toMolecular-Orhitel Theary, Holt . NewYork 1968. (12) Dewar, M.J. S., and Wor1ey.S. D., J Chem. Ph ys, 50,654 (19691.(2) Liberlea, A . , - I ~ ~ & ~ ~ ~ ~ ~oThhhhti t i~ Ma ii Chhmiitryl.. M ~ ~ ~ ~ I ~ ,~~ y Y Y k (13) These figurea are taken from Streifwieser,h. and Owens. P. H., Orbital and

1968. Electron Density Disgrsms, Maem illsn, NewYork. 1973. p. 123.(3) Streitwieser, Jr. , A , ~MoleeularOrbital m om or org ani c chemkte, wil*, ka (14) Ivash,E. V.,and Dennisan, D. M ., J Chem. P k m 21.1804 U953).

York. 1961. (15) H ehre, W.J. Stewart, R. F.,snd Pople, J. A. J Chem. Phys. 51.2657 (1969).(4) Roberta, J D.. ' Notes on M o l ~ u l a rOrh itel Cslculations: Benjamin, NewYork (16) Wiherg,K. B.. Ph ys id Olganic Chemistry, John WileyandSons ne.. New York

1961. 1964, p 66 see ootnote 23 therein).

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