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Aust.

J.

Chem., 1978,31, 199-201

Short Communicat ions

Hiickel Theory with Overlap

for the Three-Electron Bond

Richard D. Harcourt

Department of Chemistry, University of Melbourne,

Parkville, Vic. 3052.

Abstract

For a three-electron bond between two equivalent atoms, Hiickel molecular orbital theory with

overlap generates a resonance stabilization energy which is formally identical to that obtained

from molecular orbital theory when electronic repulsions are included and the Mulliken

approximation ab a +S aa+bb)) is invoked. The latter energy is the constructive interference

energy.

Huckel molecular orbital theory with atomic orbital overlap integrals (S) included

has been used1 to predict that the three-electron bond* A . B between a pair of

equivalent atoms generates destabilizing interactions when

S

>

113. Specifically, if

A and B are equivalent atoms with (normalized) overlapping atomic orbitals a and

b, then the bonding and antibonding molecular orbitals

+ ( a + b ) / ( 2 + 2 ~ ) ~ ~ nd - (a-b)/(2-2S)1 2

have Hiickel molecular orbital energies

E

(a+P)/(l + S ) and

E -

(a-P)/(l-S)

for which u and /3 are the Coulomb and resonance integrals of Hiickel theory. The

total Huckel electronic energy for the three-electron bond configuration

( +)2( -)1

is then given by equation (I), which gives

E

>

3a when

S

>

113. The energy 3a is

the energy for

A :

. B or

A .

:B when a and b do not overlap. For but not

a

orbitals,

zero overlap may be obtained by rotation of a relative to b around the AB bond

Here, we follow the usual practice to represent the three-electron bond as

A . .

.B.However, the

Green and Linnett2 representation A . B is more appropriate, because there is effectively only one

bonding electron; the antisymmetrized product wavefunction y,p+y- I with the normalized

molecular orbitals defined above, is equivalent2g3 o av,b 1 with one bonding and two non-

bonding electrons. It may be noted here that if the odd electron of y- can overlap with a singly

occupied orbital located on a third atom Y then these electrons may be spin-paired to generate4

an 'increased-valence' structure

Y-A

. B, for which three electrons participate in bonding. (The

latter electrons occupy the spin-orbitals y v- and y + , or 7,y and v+.

Baird, N. C . J . Chem. Educ., 1977, 54, 291.

Green, M. and Linnett, J. W. J Chem. Soc., 1960, 4945.

Linnett, J. W. Sci. Prog. London), 1972, 60, 1.

Harcourt, R. D., J Mol. Struct., 1972, 11, 1;

1973, 18, 515; Biopolymers, 1972, 11, 1551;

J. Phys. B, 1974,

7,

L41; see also Harcourt, R. D., and Scollary, G

R.,

Inorg. Nucl. Chem.

Lett., 1975, 11, 821, for three-electron bond theory and transition-metal complexes.

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Short Communications

Huc kel theory d oes not explicitly trea t the electron-electron interaction s.

If

these are included in the three-electron Hamiltonian operator, then both the

molecular orbital and valence-bond wavefunctions (namely

j

+

-

and

(1 aab a5b )) /(2 ~ ) ' ~ ' ) enerate the electronic energy of equation

(5)

for the

three-electron bond . In equation (5), the a an d

PC

are core Coulom b a nd resonance

integrals, and

(aa I bb)

=

(a(l)b(2)

I

llr12

I

a(l)b(2))

etc.

The electronic energy for either A: .B or A . :B, and the resonance energy

fo r a given internuclear sepa ration are then given by eq uation s (6) and (7). If we

introduce the Mulliken approximation5 for the ab orbital product (ab &S (aa+bb))

into each of (aa I ab) and (ab I ba) in equation (7), then E(resonance) reduces to

EI

=

(Po SuO) / (l S )

which is formally identical to equation (4) for the Huckel theory.

(aa I aa)

+

2

+

S)(aa I bb)

+

2(1- S)(aa

I

ab) 1 3S)(ab I ba)

(1 + S ) ( l - S 2 )

(5)

(3

S

)ac SP0 (aa I aa) 2(aa I bb) s aa I ab) ab I ba)

E(A: .B)

=

1 S2

1

S2

(6)

E(resonance) E (A . .B

E

(A : .

B)

sl 2(1+

S

)(aa I ab)

S

{(aa aa)

+

aa I bb) 2(ab I ba))

l + S

(1 + S ) ( 1 - S 2 )

(7)

For each of the ground-state molecular orbital configurations of H, +,H,, He,

an d He,, the Huc kel resonance stabilization energy* may be expressed as

n( P- Sa )/( l+ S) , with n defined as half the difference between the number

of bonding electrons and the numb er of antibo nding electrons. Th e Huckel

electronic and resonance energies for the Heitler-London wavefunction of H z

are equal to 2(x

+

SP) / ( l+

S

) and 2 S (P x)/( l+

S

).

Huckel theory predicts

th at the Heitler-London fun ctio n (1 a 6

+

bii 1)/(2

+

2 S )11' shou ld be degenerate

with the ion ic w avefunc tion (1 aii I

+

b6 1)/(2+2 s 2)112 this result of course does

no t pertain when electron-electron interactions ar e taken acco unt of.

Acknowledgments

I

thank Drs N C. Baird, J Burdett and P. Mehrotra for useful discussion, and

Professor

R

Hoffmann for the use of the facilities of his department at Cornell

University.

Manuscript received 2 August

1977

Each of the Slater determinants

a6

and

ba

for H z has a Hiickel energy of

2 ~ l

hen overlap is

included. For He the molecular orbital and atomic orbital configurations 1

v v v-ij7-

and

aab6 are equivalent and therefore no resonance stabilization is possible.