Covalent Bonding & Molecular Orbital Theory

Covalent Bonding &

Covalent Bonding &

Molecular Orbital Theory Molecular Orbital Theory

Chemistry 754 Chemistry 754 Solid State Chemistry Solid State Chemistry Dr. Patrick Woodward Dr. Patrick Woodward

Lecture #16 Lecture #16

References - MO Theory References - MO Theory

Molecular orbital theory is covered in many places Molecular orbital theory is covered in many places including most general inorganic chemistry texts.

including most general inorganic chemistry texts.

The material for this lecture (along with many of the The material for this lecture (along with many of the figures) was taken from the following two texts:

figures) was taken from the following two texts:

“Orbital Interactions in Chemistry”

“Orbital Interactions in Chemistry”

Thomas Albright, Jeremy K. Burdett &

Thomas Albright, Jeremy K. Burdett & Myung Myung- -Hwan Whangbo Hwan Whangbo, , Wiley & Sons, New York (1985).

Wiley & Sons, New York (1985).

“Chemical Bonding in Solids”

“Chemical Bonding in Solids”

Jeremy K. Burdett, Oxford University Press, Oxford (1995).

Jeremy K. Burdett, Oxford University Press, Oxford (1995).

Questions to Consider Questions to Consider

•

• Why is H Why is H _{2 2} O bent rather than linear? Why is NH O bent rather than linear? Why is NH _{3 3} pyramidal rather than planar?

pyramidal rather than planar?

• • Why are Sn Why are Sn and and Pb Pb metals, while metals, while Si Si and and Ge Ge are are semiconductors?

semiconductors?

• • Why are the Why are the π electrons π electrons delocalized delocalized in benzene (C in benzene (C _{6 6} H H _{6 6} ) ) and localized in

and localized in cyclobutadiene cyclobutadiene (C (C _{4 4} H H _{4 4} )? )?

• • In oxides, chalcogenides In oxides, chalcogenides and halides explain the and halides explain the following coordination preferences:

following coordination preferences:

–

– Cu Cu

& Mn & Mn

→ → distorted octahedral environment distorted octahedral environment –

– Ni Ni

and Fe and Fe

→ → regular octahedral environment regular octahedral environment – – Pd Pd

and Pd and Pd

→ → square planar environment square planar environment –

– Pb Pb

, Sn , Sn

, Bi , Bi

, Sb , Sb

→ → asymmetric coordination environment asymmetric coordination environment

MO Diagram for H MO Diagram for H _{2 2}

The number of MO’s is equal to the The number of MO’s is equal to the

number of atomic

number of atomic orbitals orbitals. . Each MO can hold 2 electrons (with Each MO can hold 2 electrons (with

opposite spins).

opposite spins).

The

The antibonding antibonding MO has a nodal plane MO has a nodal plane between atoms and

between atoms and ⊥ ⊥ to the bond. to the bond.

As the spatial overlap increases As the spatial overlap increases ψ ψ

(bonding MO) is stabilized and (bonding MO) is stabilized and ψ ψ

(

(antibonding antibonding MO) is destabilized. MO) is destabilized.

The destabilization of the

The destabilization of the antibonding antibonding MO is always greater than the MO is always greater than the stabilization of the bonding MO.

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.

E

1 1 ^{st st} Order MO Diagram for O Order MO Diagram for O _{2 2}

The 2s

The 2s orbitals orbitals have a lower energy have a lower energy than the 2p

than the 2p orbitals orbitals. . The

The σ σ-bonds have a greater spatial -bonds have a greater spatial overlap than the

overlap than the π π-bonds. This leads to -bonds. This leads to a larger splitting of the bonding and a larger splitting of the bonding and

antibonding orbitals antibonding orbitals. . The 2p

The 2p

and 2p and 2p

π π-interaction produces -interaction produces to two sets of degenerate

to two sets of degenerate orbitals orbitals. . The MO’s have symmetry descriptors, The MO’s have symmetry descriptors, σ σ

, , σ σ

, , π π

, , π π

within point group D within point group D

. . Mixing is allowed between MO

Mixing is allowed between MO’ ’s of the s of the same symmetry.

same symmetry.

In O

In O

there are 12 valence electrons there are 12 valence electrons and each of the 2p

and each of the 2pπ π

orbitals orbitals ( (π π

) are ) are singly occupied. Thus the bond order = singly occupied. Thus the bond order =

2, and O

2, and O

is paramagnetic. is paramagnetic.

E

2 2 ^{nd nd} Order MO Diagram for O Order MO Diagram for O _{2 2} (N (N _{2 2} ) )

A more accurate depiction of the bonding A more accurate depiction of the bonding takes into account mixing of of MO’s with takes into account mixing of of MO’s with

the same symmetry (

the same symmetry (σ σ

& & σ σ

). The ). The consequences of this 2

consequences of this 2

order effect are: order effect are:

The lower energy orbital is stabilized while The lower energy orbital is stabilized while

the higher energy orbital is

the higher energy orbital is destablized destablized. . The s and p character of the

The s and p character of the σ σ MO’s MO’s becomes mixed.

becomes mixed.

The mixing becomes more pronounced as The mixing becomes more pronounced as

the energy separation decreases.

the energy separation decreases.

E

The atomic

The atomic orbitals orbitals of the more of the more electronegative atom are lowered.

electronegative atom are lowered.

The splitting between bonding and The splitting between bonding and antibonding

antibonding MO’s now has an ionic ( MO’s now has an ionic (E E

) ) and a covalent (

and a covalent (E E

) component. ) component.

The ionic component of the splitting ( The ionic component of the splitting (E E

) )

increases as the

increases as the electronegativity electronegativity difference increases.

difference increases.

The covalency The covalency and the covalent and the covalent stabilization/destabilization decrease as stabilization/destabilization decrease as

the the electronegativity electronegativity difference difference increases.

increases.

The orbital character of the more The orbital character of the more electronegative atom is enhanced in the electronegative atom is enhanced in the

bonding MO and diminished in the bonding MO and diminished in the

antibonding antibonding MO. MO.

Heteronuclear

Heteronuclear Case & Case & Electronegativity Electronegativity

E

E _i

Linear AX

Linear AX _{2 2} (H (H _{2 2} O) MO Diagram O) MO Diagram

In linear H

In linear H

O the O 2s and O 2p O the O 2s and O 2p

orbitals

orbitals could form could form σ σ-bonds to H, -bonds to H, while the O 2p

while the O 2p

& 2p & 2p

orbitals orbitals would be non-bonding.

would be non-bonding.

Bent AX

Bent AX _{2 2} (H (H _{2 2} O) MO Diagram O) MO Diagram

In bent H

In bent H

O the O 2s σ O the O 2s σ

orbital and orbital and the O 2p

the O 2p

orbital are allowed to mix orbital are allowed to mix by symmetry, lowering the energy of by symmetry, lowering the energy of

the O 2p

the O 2p

orbital. Now there is only orbital. Now there is only one non-bonding orbital (O 2p one non-bonding orbital (O 2p

) )

Walsh Diagrams & 2

Walsh Diagrams & 2 ^{nd nd} Order JT Distortions Order JT Distortions

HOMOHOMO

Walsh Diagram Walsh Diagram

Shows how the MO levels vary as a Shows how the MO levels vary as a function of a geometrical change.

function of a geometrical change.

Walsh’s Rule Walsh’s Rule

A molecule adopts the structure A molecule adopts the structure that best stabilizes the HOMO. If that best stabilizes the HOMO. If

the HOMO is unperturbed the the HOMO is unperturbed the occupied MO lying closest to it occupied MO lying closest to it

governs the geometrical governs the geometrical

preference.

preference.

2 2 ^{nd nd} Order Jahn-Teller Dist. Order Jahn-Teller Dist.

A molecule with a small energy gap A molecule with a small energy gap

between the occupied and between the occupied and unoccupied MO’s is susceptible to a unoccupied MO’s is susceptible to a structural distortion that allows structural distortion that allows

intermixing between them.

intermixing between them.

Idealized

Idealized ββββ- ββββ -Cristobalite Cristobalite (SiO (SiO _{2 2} ) )

Space Group = Fd3m (Cubic) Space Group = Fd3m (Cubic)

Si Si-O- -O-Si Si ∠ ∠ = 180 = 180° ° sp bonding at O sp bonding at O

, , 2 nonbonding O 2p

2 nonbonding O 2p orbitals orbitals

Actual

Actual ββββ- ββββ -Cristobalite Cristobalite (SiO (SiO _{2 2} ) )

Space Group = I-42d (Tetragonal) Space Group = I-42d (Tetragonal)

Si

Si-O- -O-Si Si ∠ = 147 ∠ = 147° °

“

“sp sp

” bonding at O ” bonding at O

Covalent Bonding & the Structure of Covalent Bonding & the Structure of

Cristobalite Cristobalite

Walsh Diagram for NH Walsh Diagram for NH _{3 3}

HOMOHOMO

In the planar (D In the planar (D

) ) form the HOMO is a form the HOMO is a non-bonding O 2p non-bonding O 2p orbital (a

orbital (a

) containing ) containing 2 electrons.

2 electrons.

In the pyramidal (C In the pyramidal (C

) ) form the N 2s – H 1s form the N 2s – H 1s σ σ

orbital (a

orbital (a

) can mix ) can mix with the nonbonding O with the nonbonding O 2p orbital. Stabilizing 2p orbital. Stabilizing

the HOMO.

the HOMO.

Tetrahedral AX

Tetrahedral AX _{4 4} (CH (CH _{4 4} ) MO Diagram ) MO Diagram

Notice that while both the 2s Notice that while both the 2s

and 2p

and 2p orbitals orbitals on Carbon are on Carbon are involved in bonding, in a perfect involved in bonding, in a perfect tetrahedron mixing of the s (a tetrahedron mixing of the s (a

) )

and p (t

and p (t

) orbitals ) orbitals is forbidden. is forbidden.

Diamonds and Lead Diamonds and Lead

Structure & Properties of Structure & Properties of

the Group 14 Elements the Group 14 Elements

Element

Element Structure Structure E E

(eV ( eV) ) C

C Diamond Diamond 5.5 5.5 Si

Si Diamond Diamond 1.1 1.1 Ge Ge Diamond Diamond 0.7 0.7 α α- -Sn Sn Diamond Diamond 0.1 0.1 β β- -Sn Sn Tetragonal Metal Tetragonal Metal Pb

Pb FCC FCC Metal Metal As you go proceed down the group As you go proceed down the group

the tendency for the s-

the tendency for the s-orbitals orbitals to become involved in bonding to become involved in bonding diminishes. This destabilizes diminishes. This destabilizes tetrahedral coordination and tetrahedral coordination and

semiconducting

semiconducting/insulating /insulating behavior.

behavior.

2p 2p 2s 2s

2p 2p 2s 2s a

a

a a

t t

t t

C C

6p 6p

6s 6s a

a

a a

t t

t t

Pb Pb

6p 6p

6s 6s

2 2 ^{nd nd} Order JT Distortion in PbO Order JT Distortion in PbO

Pb Pb 6s 6s HOMO HOMO

In both polymorphs of

In both polymorphs of PbO PbO (red (red PbO PbO, the tetragonal form is shown above) the Pb , the tetragonal form is shown above) the Pb

ions adopt a very asymmetric coordination environment. The driving force for this ions adopt a very asymmetric coordination environment. The driving force for this

is to lower the energy of the filled,

is to lower the energy of the filled, antibonding Pb antibonding Pb 6s 6s orbitals orbitals, by mixing with an , by mixing with an empty

empty Pb Pb 6p orbital. Such mixing is forbidden by symmetry in tetrahedral and 6p orbital. Such mixing is forbidden by symmetry in tetrahedral and octahedral coordination, so a distortion to a lower symmetry leading to the octahedral coordination, so a distortion to a lower symmetry leading to the

formation of the so-called “

formation of the so-called “stereoactive stereoactive electron lone pair” occurs. Such electron lone pair” occurs. Such distortions are common for main group ions with their valence s electrons (

distortions are common for main group ions with their valence s electrons (Tl Tl

, Bi , Bi

, , Sn

Sn

, Sb , Sb

, etc.). This distortion is similar to the one seen in NH , etc.). This distortion is similar to the one seen in NH

. .

Cyclic

Cyclic Polyenes Polyenes

Benzene (C Benzene (C _{6 6} H H _{6 6} ) )

Cyclobutadiene

Cyclobutadiene (C (C _{4 4} H H _{4 4} ) ) E

Consider two cyclic Consider two cyclic C C

H H

systems. The sketches to the systems. The sketches to the left show the phases of the C left show the phases of the C 2p 2p

orbitals orbitals that are responsible that are responsible

for π for π-interactions. In each -interactions. In each system there are n

system there are n π π-MO’s. The -MO’s. The lowest energy orbital has no lowest energy orbital has no nodes (all

nodes (all orbitals orbitals in phase) while in phase) while the highest energy state has the the highest energy state has the

maximum number (n/2).

maximum number (n/2).

In C

In C

H H

there is a large HOMO- there is a large HOMO- LUMO gap and the e

LUMO gap and the e

orbitals orbitals are fully occupied.

are fully occupied.

In C In C

H H

the the e e

orbital HOMO is orbital HOMO is

½ occupied (triplet ground

½ occupied (triplet ground state).

state).

1 1 ^{st st} Order Jahn-Teller Distortion in C Order Jahn-Teller Distortion in C _{4 4} H H _{4 4}

E

In practice

In practice cyclobutadiene cyclobutadiene does not does not form a regular square (D

form a regular square (D

), but ), but undergoes a distortion to a undergoes a distortion to a rectangular shape (D

rectangular shape (D

). This ). This stabilizes one of the

stabilizes one of the HOMO’s HOMO’s (which (which becomes doubly occupied) and becomes doubly occupied) and destabilizes the other (which destabilizes the other (which

becomes empty).

becomes empty).

This leads to formation of two This leads to formation of two localized double bonds. Hence, C localized double bonds. Hence, C

H H

is said to be

is said to be antiaromatic antiaromatic. .

1 1 ^{st st} Order Jahn-Teller Dist. Order Jahn-Teller Dist.

A non-linear molecule with an A non-linear molecule with an incompletely filled degenerate incompletely filled degenerate HOMO is susceptible to a structural HOMO is susceptible to a structural

distortion that removes the distortion that removes the

degeneracy.

degeneracy.

Octahedral Coordination Octahedral Coordination

The diagram to the left shows a MO The diagram to the left shows a MO

diagram for a transition metal diagram for a transition metal octahedrally coordinated by

octahedrally coordinated by σ σ-bonding -bonding ligands

ligands. ( . (π π-bonding has been neglected) -bonding has been neglected) Note that in an octahedron there is no Note that in an octahedron there is no

mixing between s, p and d-

mixing between s, p and d-orbitals orbitals. . For a main group metal the same For a main group metal the same diagram applies, but we neglect the d- diagram applies, but we neglect the d-

orbitals orbitals. . The t

The t

orbitals orbitals ( (d d

, ,d d

,d , d

) are π ) are π- - antibonding

antibonding (not shown), while the (not shown), while the e e

orbitals orbitals (d (d

,d ,d

) are σ ) are σ- - antibonding

antibonding. The latter are higher . The latter are higher in energy since the spatial overlap in energy since the spatial overlap

of the

of the σ σ-interaction is stronger. -interaction is stronger.

Square Planar Coordination Square Planar Coordination

The diagram to the left shows a MO The diagram to the left shows a MO diagram for a transition metal in square diagram for a transition metal in square planar coordination. (

planar coordination. ( π-bonding has been π -bonding has been neglected)

neglected)

Among the changes the most important is Among the changes the most important is

that now the s and d

that now the s and d

orbitals orbitals can mix, can mix, which stabilizes the d

which stabilizes the d

and removes the and removes the degeneracy of the

degeneracy of the e e

orbitals orbitals. . Transition metals with electron counts that Transition metals with electron counts that lead to partially filled

lead to partially filled e e

orbitals orbitals (HS d (HS d

, d , d

& d

& d

in particular) will be prone to undergo in particular) will be prone to undergo distortions from octahedral toward square distortions from octahedral toward square

planar.

planar.

The d

The d

ions Pd ions Pd

and Pt and Pt

have a strong have a strong preference for sq. planar coordination, but preference for sq. planar coordination, but with Ni

with Ni

the crystal field splitting is usually the crystal field splitting is usually too small to overcome the spin pairing too small to overcome the spin pairing energy and octahedral coordination results.

energy and octahedral coordination results.

Jahn-Teller Distortions:

Jahn-Teller Distortions:

The long and the short of it.

The long and the short of it.

The Jahn-Teller theorem tells The Jahn-Teller theorem tells us there should be a distortion us there should be a distortion

when the

when the e e

orbitals orbitals of a TM of a TM octahedral complex are partially octahedral complex are partially occupied, but it doesn’t tell us occupied, but it doesn’t tell us what type of distortion should what type of distortion should occur.

occur. To a first approximation To a first approximation two choices give the same two choices give the same energetic stabilization.

energetic stabilization.

2 long + 4 short bonds – 2 long + 4 short bonds – stabilizes the d

stabilizes the d

orbital orbital 2 short + 4 long bonds – 2 short + 4 long bonds – stabilizes the d

stabilizes the d

orbital orbital .

.

Distortions in d

Distortions in d ^{9 9} & d & d ^{10 10} Halides Halides

In practice Cu

In practice Cu

(d (d

) and Mn ) and Mn

(HS) almost always take the 2 long + 4 (HS) almost always take the 2 long + 4 short

short distortion distortion, and the distortions are usually considerably larger , and the distortions are usually considerably larger with Cu

with Cu

. In contrast d . In contrast d

ions, such as Hg ions, such as Hg

adopt very large adopt very large 2 short + 2 short + 4 long distortions

4 long distortions (in many cases the distortion is so large that the (in many cases the distortion is so large that the coordination is essentially linear). For example consider the bond coordination is essentially linear). For example consider the bond distances in CuBr

distances in CuBr

(4 × (4 × 2.40 2.40Å Å, , 2 2 × × 3.18 3.18Å Å) and HgBr ) and HgBr

(4 (4 × × 3.23 3.23Å Å, , 2 2 × × 2.48

2.48Å Å), both of which adopt distorted CdI ), both of which adopt distorted CdI

structures. structures. Why is this Why is this so? Why do d

so? Why do d

ions distort at all? ions distort at all?

Short bonds drawn with solid lines.

Long bonds drawn with dotted lines.

Jahn-Teller Distortions d

Jahn-Teller Distortions d _{z2 z2} -s Mixing -s Mixing

The empty

The empty n n s orbital is of s orbital is of appropriate symmetry to mix with appropriate symmetry to mix with

the (

the ( n-1 n-1 )d )d

orbital, but not with orbital, but not with the (

the ( n-1 n-1 )d )d

orbital. This orbital. This dictates the details of the dist.

dictates the details of the dist.

d

d

case (Cu case (Cu

): The d ): The d

-s mixing -s mixing favors preferential occupation of favors preferential occupation of

the d

the d

orbital (2 long + 4 short orbital (2 long + 4 short favored)

favored) d

d

case (Hg case (Hg

): The d ): The d

-s mixing -s mixing is largest when the energy is largest when the energy separation between the two is separation between the two is minimized (

minimized ( ∆∆∆∆ ∆∆∆∆E E

> > ∆∆∆∆ ∆∆∆∆E E

). (2 short + ). (2 short + 4 long favored)

4 long favored)