Covalent Bonding &
Molecular Orbital Theory
Chemistry 754
Solid State Chemistry
Dr. Patrick Woodward
Lecture #16
References - MO Theory
Molecular orbital theory is covered in many places
including most general inorganic chemistry texts.
The material for this lecture (along with many of the
figures) was taken from the following two texts:
“Orbital Interactions in Chemistry”
Thomas Albright, Jeremy K. Burdett & MyungMyung-Hwan Whangbo,
Whangbo,
Wiley & Sons, New York (1985).
“Chemical Bonding in Solids”
Jeremy K. Burdett, Oxford University Press, Oxford (1995).
Questions to Consider
• Why is H2O bent rather than linear? Why is NH3
pyramidal rather than planar?
• Why are Sn and Pb metals, while Si and Ge are
semiconductors?
• Why are the π electrons delocalized in benzene (C6H6)
and localized in cyclobutadiene (C4H4)?
• In oxides, chalcogenides and halides explain the
following coordination preferences:
–
–
–
–
Cu2+ & Mn3+ → distorted octahedral environment
Ni2+ and Fe3+ → regular octahedral environment
Pd2+ and Pd2+ → square planar environment
Pb2+, Sn2+, Bi3+, Sb3+ → asymmetric coordination environment
MO Diagram for H2
The number of MO’s is equal to the
number of atomic orbitals.
orbitals.
E
Each MO can hold 2 electrons (with
opposite spins).
The antibonding MO has a nodal plane
between atoms and ⊥ to the bond.
As the spatial overlap increases ψ1
(bonding MO) is stabilized and ψ2
(antibonding MO) is destabilized.
The destabilization of the antibonding
MO is always greater than the
stabilization of the bonding MO.
In the diagrams at the top and bottom the solid line
denotes the electron density from MO theory and the
dashed line the electron density from superimposing to
atomic orbitals.
1st Order MO Diagram for O2
The 2s orbitals have a lower energy
than the 2p orbitals.
orbitals.
The σ-bonds have a greater spatial
overlap than the π-bonds. This leads to
a larger splitting of the bonding and
antibonding orbitals.
orbitals.
E
The 2px and 2py π-interaction produces
to two sets of degenerate orbitals.
orbitals.
The MO’s have symmetry descriptors,
σg+, σu+, πg, πu within point group D∞h.
Mixing is allowed between MO’
MO’s of the
same symmetry.
In O2 there are 12 valence electrons
and each of the 2pπ
2pπ* orbitals (πg) are
singly occupied. Thus the bond order =
2, and O2 is paramagnetic.
2nd Order MO Diagram for O2 (N2)
E
A more accurate depiction of the bonding
takes into account mixing of of MO’s with
the same symmetry (σ
(σg+ & σu+). The
nd
consequences of this 2 order effect are:
The lower energy orbital is stabilized while
the higher energy orbital is destablized.
destablized.
The s and p character of the σ MO’s
becomes mixed.
The mixing becomes more pronounced as
the energy separation decreases.
Heteronuclear Case & Electronegativity
The atomic orbitals of the more
electronegative atom are lowered.
E
The splitting between bonding and
antibonding MO’s now has an ionic (E
(Ei)
and a covalent (E
(Ec) component.
Ei
The ionic component of the splitting (E
(Ei)
increases as the electronegativity
difference increases.
The covalency and the covalent
stabilization/destabilization decrease as
the electronegativity difference
increases.
The orbital character of the more
electronegative atom is enhanced in the
bonding MO and diminished in the
antibonding MO.
Linear AX2 (H2O) MO Diagram
In linear H2O the O 2s and O 2pz
orbitals could form σ-bonds to H,
while the O 2px & 2py orbitals
would be non-bonding.
Bent AX2 (H2O) MO Diagram
In bent H2O the O 2s σ∗ orbital and
the O 2px orbital are allowed to mix
by symmetry, lowering the energy of
the O 2px orbital. Now there is only
one non-bonding orbital (O 2py)
Walsh Diagrams & 2nd Order JT Distortions
Walsh Diagram
Shows how the MO levels vary as a
function of a geometrical change.
Walsh’s Rule
HOMO
A molecule adopts the structure
that best stabilizes the HOMO. If
the HOMO is unperturbed the
occupied MO lying closest to it
governs the geometrical
preference.
2nd Order Jahn-Teller Dist.
A molecule with a small energy gap
between the occupied and
unoccupied MO’s is susceptible to a
structural distortion that allows
intermixing between them.
Covalent Bonding & the Structure of
Cristobalite
Idealized β-Cristobalite (SiO2)
Space Group = Fd3m (Cubic)
Si-OSi ∠ = 180°
Si-O-Si
180°
sp bonding at O2-,
2 nonbonding O 2p orbitals
Actual β-Cristobalite (SiO2)
Space Group = I-42d (Tetragonal)
Si-OSi ∠ = 147°
Si-O-Si
147°
2
“sp ” bonding at O2-
Walsh Diagram for NH3
In the planar (D3h)
form the HOMO is a
non-bonding O 2p
orbital (a2) containing
2 electrons.
HOMO
In the pyramidal (C3v)
form the N 2s – H 1s σ*
orbital (a1) can mix
with the nonbonding O
2p orbital. Stabilizing
the HOMO.
Tetrahedral AX4 (CH4) MO Diagram
Notice that while both the 2s
and 2p orbitals on Carbon are
involved in bonding, in a perfect
tetrahedron mixing of the s (a1)
and p (t2) orbitals is forbidden.
C
a1*
2p
2s
Diamonds and Lead
t2*
Structure & Properties of
the Group 14 Elements
2p
2s
t2
a1
Pb
6p
6s
t2*
6p
t2
a1*
a1
6s
Element
C
Si
Ge
α-Sn
β-Sn
Pb
Structure
Diamond
Diamond
Diamond
Diamond
Tetragonal
FCC
Eg(eV)
eV)
5.5
1.1
0.7
0.1
Metal
Metal
As you go proceed down the group
the tendency for the s-orbitals
s-orbitals
to become involved in bonding
diminishes. This destabilizes
tetrahedral coordination and
semiconducting/insulating
semiconducting/insulating
behavior.
2nd Order JT Distortion in PbO
Pb 6s
HOMO
In both polymorphs of PbO (red PbO,
PbO, the tetragonal form is shown above) the Pb2+
ions adopt a very asymmetric coordination environment. The driving force for this
is to lower the energy of the filled, antibonding Pb 6s orbitals,
orbitals, by mixing with an
empty Pb 6p orbital. Such mixing is forbidden by symmetry in tetrahedral and
octahedral coordination, so a distortion to a lower symmetry leading to the
formation of the so-called “stereoactive
“stereoactive electron lone pair” occurs. Such
distortions are common for main group ions with their valence s electrons (Tl
(Tl+, Bi3+,
Sn2+, Sb3+, etc.). This distortion is similar to the one seen in NH3.
Cyclic Polyenes
Benzene (C6H6)
E
Consider two cyclic CnHn
systems. The sketches to the
left show the phases of the C
2pz orbitals that are responsible
for π-interactions. In each
system there are n π-MO’s. The
lowest energy orbital has no
nodes (all orbitals in phase) while
the highest energy state has the
maximum number (n/2).
In C6H6 there is a large HOMOLUMO gap and the e1g orbitals
are fully occupied.
In C4H4 the eg orbital HOMO is
½ occupied (triplet ground
state).
Cyclobutadiene (C4H4)
1st Order Jahn-Teller Distortion in C4H4
In practice cyclobutadiene does not
form a regular square (D4h), but
undergoes a distortion to a
rectangular shape (D2h). This
stabilizes one of the HOMO’s (which
becomes doubly occupied) and
destabilizes the other (which
becomes empty).
E
This leads to formation of two
localized double bonds. Hence, C4H4
is said to be antiaromatic.
antiaromatic.
1st Order Jahn-Teller Dist.
A non-linear molecule with an
incompletely filled degenerate
HOMO is susceptible to a structural
distortion that removes the
degeneracy.
Octahedral Coordination
The diagram to the left shows a MO
diagram for a transition metal
octahedrally coordinated by σ-bonding
ligands.
ligands. (π
(π-bonding has been neglected)
Note that in an octahedron there is no
mixing between s, p and d-orbitals
d-orbitals..
For a main group metal the same
diagram applies, but we neglect the dorbitals.
orbitals.
The t2g orbitals (d
(dxy,dyz,dxz) are πantibonding (not shown), while the
eg orbitals (dz2,dy2-y2) are σantibonding.
antibonding. The latter are higher
in energy since the spatial overlap
of the σ-interaction is stronger.
Square Planar Coordination
The diagram to the left shows a MO
diagram for a transition metal in square
planar coordination. (π
(π-bonding has been
neglected)
Among the changes the most important is
that now the s and dz2 orbitals can mix,
which stabilizes the dz2 and removes the
degeneracy of the eg orbitals.
orbitals.
Transition metals with electron counts that
lead to partially filled eg orbitals (HS d4, d8
& d9 in particular) will be prone to undergo
distortions from octahedral toward square
planar.
The d8 ions Pd2+ and Pt2+ have a strong
preference for sq. planar coordination, but
with Ni2+ the crystal field splitting is usually
too small to overcome the spin pairing
energy and octahedral coordination results.
Jahn-Teller Distortions:
The long and the short of it.
The Jahn-Teller theorem tells
us there should be a distortion
when the eg orbitals of a TM
octahedral complex are partially
occupied, but it doesn’t tell us
what type of distortion should
occur. To a first approximation
two choices give the same
energetic stabilization.
2 long + 4 short bonds –
stabilizes the dz2 orbital
2 short + 4 long bonds –
stabilizes the dx2-y2 orbital
.
Distortions in d9 & d10 Halides
Short bonds drawn with
solid lines.
Long bonds drawn
with dotted lines.
In practice Cu2+ (d9) and Mn3+ (HS) almost always take the 2 long + 4
short distortion,
distortion, and the distortions are usually considerably larger
with Cu2+. In contrast d10 ions, such as Hg2+ adopt very large 2 short +
4 long distortions (in many cases the distortion is so large that the
coordination is essentially linear). For example consider the bond
distances in CuBr2 (4 × 2.40Å
2.40Å, 2 × 3.18Å
3.18Å) and HgBr2 (4 × 3.23Å
3.23Å, 2 ×
2.48Å
2.48Å), both of which adopt distorted CdI2 structures. Why is this
so? Why do d10 ions distort at all?
Jahn-Teller Distortions dz2-s Mixing
The empty ns orbital is of
appropriate symmetry to mix with
the (n-1)dz2 orbital, but not with
the (n-1)dx2-y2 orbital. This
dictates the details of the dist.
d9 case (Cu2+): The dz2-s mixing
favors preferential occupation of
the dz2 orbital (2 long + 4 short
favored)
d10 case (Hg2+): The dz2-s mixing
is largest when the energy
separation between the two is
minimized (∆
(∆E2 > ∆E1). (2 short +
4 long favored)