Zumdahl’s Chapter 9
Covalent Bonding
Orbitals
Chapter Contents
Hybridization and LE
dnspm
Molecular Orbitals
Bond Order
Bonding in Homonuclear
Diatomics
Paramagnetism
Heteronuclear Bonding
LE + MO
Atomic Orbital Hybridization
VSEPR postulates repulsion geometries, but are
atomic wavefunctions flexible enough to supply?
A key to wave mechanics is superposition,
creating new waves from interference of old ones.
Degenerate (same energy) wavefunctions can mix
arbitrarily to give new degenerate sets. #  #
If energy lowering is possible, even near-degenerate
(similar energy) sets of  can mix to new sets.
The Joys of Promotion
Minimization of electron repulsion is the motive
for mixing a set of AOs (Atomic Orbitals) that
produces new AOs in the VSEPR directions.
In addition, paired electrons in AOs are already
satisfied and need no stinking bonds, but if these
partners are split up, more bonding is possible,
at the bargain expense of promotion to higher
energy orbitals. Extra bond energies pay it back.
Carbon Wins Big
The electronic configuration of carbon is
[He] 2s2 2p2 or [He] [] [ ] [ ] [ ]
As such, only CH2 or two valence bonds are
possible for carbon. Not for long.
Using the energy accessible from bond
formation, carbon promotes an s e– to a p orbital
[He] 2s1 2p3 or [He] [ ] [ ] [ ] [ ]
Making CH4 and other four valence molecules.
Blending the l out of it.
Promotion is necessary but not sufficient; we
must still mix s+px+py+pz in ways like VSEPR.
The electron density follows the new orbital
directions and the nuclei obey the bonding geometry.
We needn’t blend all of the available orbitals!
We can mix spx and leave py and pz for lone pair.
That would give us linear XAX bonding.
And that results from constructive interference in .
Wave Mixture Geometries
2s
+
Bonding Direction
2px
=
Node shifted
off x=0 plane.
sp Hybridization
Starting with two AOs, mixing must generate no
more than nor no fewer than two hybrids.
The s + px combination points up the x-axis while
the s – px combination points down the x-axis.
Together they give linear XAX molecules (BeF2).
Also, the sp hybrid is elongated in its bonding
direction for better penetration and lower energy.
Trigonal Planar Hybridization, sp2
and
mixed
with
and 2 other
combinations
gives
2
sp
Hybridization
As with sp, the leftover p orbital(s) are available
for electrons as either lone pairs or  bonds.
In contrast, the single bonds created with the
hybrids are called  bonds, where  is the
Greek s and  is the Greek p.
All 3 sp2 together look like:
Truly 120° apart
If your browser supports VRML, try http://www.chm.davidson.edu/vrml/ao/ for details of hybrid orbitals.
sp3 Tetrahedral Hybridization
The sp2 are even more penetrating than the sp
but less so than sp3. The more the p character,
the more directional the hybrid. Duhh.
Carbon can hybridize sp3, but so can N and O;
the difference is in how many hybrids have only
one electron. Those bond; the others lone pair.
The four sp3 together look like:
And the angle is now
cos–1(–1/3) = 109.43°
Hybridization Beyond
n
sp
The sp3 hybrid orbitals permit a valence of up to
4 and the expected octet.
Violating the octet demands incorporation of d
orbitals as in dsp3, d2sp3, etc.
and dx²–y²
They use dz²
Trigonal Bipyramidal
3
dsp
Mixing dz² with sp3 gives five orbitals, two of
which are axial and three equatorial.
Only one each of the two kinds are shown here, but
the other axial is just down the Z while the other two
equatorials are around Z at 120° to the one shown.
Octahedral d²sp³
Mixing dx²–y² with the five dsp³ gives the six
identical d²sp³ orbitals in Cartesian directions.
This picture is merely diagrammatic; it is not an
accurate representation of the d²sp³ wavefunctions. (It is a group of spheres and cones.)
Still, if your browser is equipped with VRML, you
can play with my first 3d world construction at
http://www.utdallas.edu/~parr/chm1315/d2sp3.wrl
How Chemists Use Hybrids
Hybrid orbitals are great shorthand notations for
building up a molecule’s geometry center by
center … part of every molecular model kit.
The skeletal structure so developed is called the
 skeleton because in-line overlap of adjacent
bonding hybrids are cylindrically symmetrical
about the internuclear axis and thus have no
(axial) angular momentum (like s for an atom).
 and the remainders
The  skeleton sets hybridization for each of the
molecule’s central atoms.
Bonded to 3 things? No lone pair? You are sp2!


I not only expect 120° bonds from you but also an
unhybridized p orbital  to that plane.
Since the rest of your octet isn’t a lone pair, you must be
 bonding to one or more of your partners.
Bonded to 3 with a lone pair? You’re sp3!

I expect 109.43° bond angles or there abouts.
 bonds? Piece of cake.
Unhybridized p orbitals on adjacent bonding
centers can overlap (inefficiently) sideways.
Density not cylindrically symmetric (like ) but does
allow for buildup between nuclei (off line-of-centers).
 bonds weaker than  but add 1 to the bond order.
Off axis, they’re vulnerable to chemical attack; 
bonds are reactive while  are relatively inert.

“Unsaturated” fats have ’s to permit metabolic degrade.
Vitamin B12
sp CN ligand
sp2 C=O
d2sp3 Co
sp3
sp3 SO4
Vitamin B12 with
its multiple bonds
Molecular Orbital (MO) Theory
Electrons don’t ignore all other nuclei beyond the
adjacent bonding pair. They’re really global.
Instead of building molecules atom by atom,
we’ll pour electrons onto a nuclear skeleton.
Hess assures us that when all the electrons are
finally present, the (binding) energies will be the
same either way.
So how do electrons respond differently this way?
Add electrons to proton framework
They see a wavefunction that spans molecule!
First approximation model to that is LCAO,
Linear Combination of Atomic Orbitals.
Study the diatomics for simplicity. The advantages
to MO will become apparent even there.
Thus, H2’s MOs are LC of 1sA and 1sB where A and
B are the labels for the two hydrogen atoms.
1s  1sA + 1sB while 1s*  1sA – 1sB (2 in; 2 out)
1s H2
A
MO also builds
density between
bonded nuclei.
B
Fortunately,
this MO holds
both electrons.
See http://www.chm.davidson.edu/vrml/mo/h2/h2.html
1s* H2 MO
A
Electron density
vacates region
between nuclei!
B
Any electron in
this antibonding
MO reduces BO
by ½.
Bond Order (BO) in MO
MO’s come in constructive (e.g., 1s) and
destructive (e.g., 1s*) combinations as regards
the internuclear region.
Since they must mimic lone pairs as well, there are
nonbonding MO’s, but they do not influence BO.
BO = ½ ( electrons MO –  electrons MO* )
First surprising consequence: H2+ has BO = ½
A stable one-electron bond is possible.
Correlates with Diss. E. and negatively with Bond R.
Using p AO’s
If internuclear axis is Z, then 2pZA – 2pZB binds
and is called 2pz. The “+” combo anti*binds.
More interesting are PX and PY which combine
off the internuclear axis as  MO’s.
2pX*  2pXA – 2pXB for example.
Note that some combinations are meaningless
because they do not overlap in bonding regions:
e.g., 2pXA + 2pYB produces no MO.
Degenerate MO’s
The energies of 2pX and 2pY are identical.
Hund’s Rule applies to MO’s just as it did AO’s.
1s2 1s*2 2s2 2s*2 2pZ2 2pX1 (C2+)
is followed by
1s2 1s*2 2s2 2s*2 2pZ2 2pX1 2pY1 (C2)
Implying not only C=C but also   or a
paramagnetic C2 diatom.
Decline of Bond Order
The pinnacle of A2 (2nd row generic diatom)
comes at N2 with electronic configuration:
[Be2] 2pZ2 2pX2 2pY2 and bond order 3, NN.
We’ve run out of bonding MO’s of the 2 shell.
Starting with O2, antibonding highest occupied
MO’s (HOMO) diminish BO.
[Be2] 2pZ2 2pX2 2pY2 2pX*1 2pY*1
Only bond order 2 but paramagnetic.
See http://www.chem.technion.ac.il/ElBookOrb/molecule.htm
Formic Acid, HCO2H
Lewis Structure would have resonance in the
conjugate base with the C-O bonds at 1.5 order.
MO generates this naturally by mixing 2pX from
both oxygens and the carbon to create:
An example of the
delocalized nature
of  bonding.
 bonding is better described as local.
Their mixing generates these double bonds.
Benzene, a textbook delocalization
After hybridizing sp2 for
the  skeleton,

Yes, you can build MO’s
from hybridized AO’s.
The 6 leftover pX orbitals
mix to give global MO’s 
the plane of the nuclei.
Before mixing they are:
Evidence of delocalized electrons comes from benzene’s magnetic “ring current.”
C6H6  Bonding
After mixing, six new
MO’s arise, 3 bonding
and 3 antibonding.
Best case at top, and
worst case at bottom.
The 6 electrons from
each carbon’s p pair up
in the 3 bonding ’s.
Knowing Where the Electrons are is
POWER
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