The Relative Energies of Molecular Orbitals for Second-Row
Homonuclear Diatomic Molecules
The Effect of s-p Mixing
Albert Haim
State University of New York, Stony Brook. NY 11794
The molecular orbital (MO) treatment of bonding is now
well-ingrained in introductory chemistry courses withmost
elementary textbooks presenting MO diagrams for the
second-row homonuclear diatomic molecules. Two diagrams are usually presented: one for the later elements
Atomic
Orbitals
Molecular
Orbitals
Atomic
Orbitals
(oxygen and fluorine) and one for the earlier elements
(boron through nitrogen). Unfortunately, the later diagrams are almost invariably incorrect. The purpose of the
present contribution is to point out the error and to rectify
it.
Atomic
Orbitals
Molecular
Orbitals
Atomic
Orbitals
Figure 1. Relative energies of 2s and 2p AO's and of resulting a and n: MO's. (a) Negligible s-p, mixing, correct diagram. (b) Substantial s-p,
mixing, incorrect diagram (see text).
Volume 68 Number 9 September 1991
737
Negligible
s-pz mixing
Substantial
s-pz mixing
Figure 2. Effect of s-p, mixing on relative energies of MO's, correct
diagram. (a)Negligibles-p,mixing. (b)Substantials-p,mixing. When
mixing is important, some bonding and antibonding relationships are
poorly defined. Then it is preferable to delete the b and 'superscripts
and retain only the symmetry significant-symbols:a, n, g, u.
MO's are obtained by linear combination of atomic orbitals (LCA07s).For the second-row elements, the AO's of
interest are Zs, Zp,, 2p,, and Zp,. These combine to yield
eight MO's designated a2(2s), au*(2s), a,b(Zp,), a.*(Zp,),
a,b(2p,), %b(2pY),ag*(2p,), and ag*(2p,). The a and a designations apply to MO's which are symmetric and antisymmetric, respectively, in rotation about the internuclear
axis (the z coordinate). The g and u subscripts refer to
symmetric and antisymmetricMO's, respectively, for inver-
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-
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-
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'De Kock, R. L.; Gray, H. B. ~hernicaj~tructure
and Bonding;
Benjamin-CumminsCo: Menlo Park, CA, 1980;p 227.
2Douglas, B.; McDaniel, D. H.; Alexander, J .J. Concepts and
ModelsoflnorganicChemisty, 2nd ed;Wiley: New York, 1983;p 135.
738
Journal of Chemical Education
sion about the center of the molecule. The b and * superscripts indicate whether the MO is bonding or antibonding
(with concentration or depletion of electron density in the
internuclear reeion. resoectivelv). Finallv. the A 0 used in
the construction of the MO is &en in pa;knthesis.
Bonding MO's constructed h m 2s and 2p, AO's have the
same symmetry (a,). Similarly, the corresponding antibondine MO's also have the same svmmetrv
" (a,.).
. ". Afundamentaraxiom of quantum theory ;s that orbitals of the
same smmetrv will mix. The extent of mixine d e ~ e n d on
s
the refative energies of the orbitals. With a Tar& energy
difference. mixing is nedieible. but it becomes imoortant
as the energies oflthe o r h k s come closer together:^^ the
relative energies of s and D, AO's varv across the oeriodic
table,'sodoeHthe extent of&ingbet&een&?s) &da(o(2)
MO's. Thus, two different MO diagrams are obtained. One
diagram applies to O2and F2whepe mixing is unimportant
(large difference in energy between 2s and 2p, AO's), and
the other applies to BP,C2,and N2where mixingis substantial (smaller differencein energy between 2s and 2pzAO's).
The diagrams usually encountered in textbooks are given
in Figure l a (no s-p, mixing) and l b (s-p, mixing). Figure
l a shows that the bondine and antibondine
are svm" MO's
~~~-~
metrically placed above agd below the component ~O's.?he
enerw
-" relationshios between AO's and MO's deoicted in
Figure l a are given correctly in most textbooks and are
based on the zero-overlap approximati~n.~
The problem arises when s-p, mixing is included. The
diagram given in most textbooks is shown in Figure lb.
Apparently, the diagram for no s-p, mixing (Fig. l a ) bas its
a2(2pZ)and n,b(2p,), a,b (2py levels interchanged. There
are two problems with the resulting incorrect diagram.
First, symmetry considerations require that rr MO's be
unaffected by s-p, mixing; only a MO's change in energy.
Second, by lowering the level of the bonding nb MO's and
leaving the a* MO's unchanged, when compared with the
2pzand 2p,AO's, the bondingx MO's are lowered more than
the antibondingrr MO's are raised. This energyrelationship
is, of course, incorrect. With the zero-overlap approximation, bonding and antibonding MO's are lowered and raised
equally? If overlap is included, the antibonding MO is
raised more than the bonding MO is lowe~ed.~
The bonding
MO will never be lowered more than the antibonding MO
is raised. The correct effect of s-p, mixing on the energies
of the MO's (zero-overlapapproximation) is given in Figure
2: a,b(2s) and aU*(2s)MO's are lowered; a2(2pz) and
am*(2p,)are raised; and rr orbitals are unchanged.
Acknowledgment
The author is grateful to Professor Joe Lauher for
illuminating discussions and critical reading of several
drafts of the manuscript.