Molecular Orbital Theory

General Chemistry

Molecular Orbital Theory

Dr. S. K. Tan

Topics to be Covered in this Lecture

Concepts of Molecular Orbitals

Correlation Energy Diagram

Prediction of Reactivity of Molecules

Bonding Theories

We have already been introduced to one of the main bonding theories -

The Valence Bond Theory(VB)

* Pairs of electrons repel each other, therefore orient themselvesPairs of electrons repel each other, therefore orient themselvesin a in a

way that minimizes these repulsions. way that minimizes these repulsions.

* Covalent bonds are formed by overlapping of atomic Covalent bonds are formed by overlapping of atomic orbitalsorbitals..

* The electrons that form a bond between two atoms are localized The electrons that form a bond between two atoms are localized

between the atoms.between the atoms.

There is however another contender to the throne for bonding theory

- and this is called The Molecular Orbital Theory (MO )

* Atomic Atomic orbitalsorbitalsare combined into new are combined into new orbitalsorbitals, called molecular , called molecular

orbitalsorbitals(MOs). (MOs).

* MOs need not be localized between two atoms. MOs need not be localized between two atoms.

* The number of MOs formed is equal to the sum of atomic The number of MOs formed is equal to the sum of atomic orbitalsorbitalsin in

all the atoms comprising the molecule.all the atoms comprising the molecule.

* When atomic When atomic orbitalsorbitalscombine to form bonding MOs, the resulting combine to form bonding MOs, the resulting

MO is lower in energy. MO is lower in energy.

* AntibondingAntibondingMOs are less stable than the MOs are less stable than the AOsAOs..

Bonding Theories

Molecular Orbital Theory

Here, if we have two nuclei that are close to each other, when we

include the electrons into consideration, the electrons will move into

molecular orbitals.

This is analogous to the electrons occupying atomic orbitals in the case

of the atoms.

For the atomic orbitals, there are s, p, d, f,... orbitals.

In the case of molecular orbitals, there are σ, π, δ, ...; where the

orbitals are determined by quantum numbers.

Molecular Orbital Theory (2)

To obtain the molecular orbital, the Schrödinger equation must be

solved, and here lies the problem.

The equation cannot be solved exactly and therefore certain

approximations must be made.

One of them involves the BornBorn--Oppenheimer ApproximationOppenheimer Approximation,

which assumes that nuclei moves much more slowly than electrons(also known as the Clamped-Nuclei Approximation).

This separates the situation into two problems - an electronic and a

nuclear motion problem.

Molecular Orbital Theory (3)

The electronic problem solves for the wave function of the electrons

while the nuclear motion solves for the motion of the nuclei.

Another approximation that is made is that the molecular orbitals may

be approximated by taking the Linear Combination of Atomic Linear Combination of Atomic

OrbitalsOrbitals ((LCAOLCAO ).).

The rationale is that most of the time the electrons are close to nuclei

and will most probably be ‘controlled’ by one of the two nuclei.

Hence, the molecular orbital will most probably be a combination of

the atomic orbitals.

Molecular Orbital Theory (4)

So, if we have two atomic orbitals, ΨA

and ΨB, we can combine them

to obtain two molecular orbitals.

Ψb= Ψ

A+ Ψ

B; and Ψ

a= Ψ

A- Ψ

B, where Ψ

bis a bonding molecular

orbital and is an Ψaanti-bonding molecular orbital.

The electron distribution is then calculated by taking the square of

that function.

Hence, Ψb

2 = ΨA

2 + 2ΨAΨ

B+ Ψ

B

2 , while the corresponding electron

distribution for the anti-bonding molecular orbital is Ψa

2 = ΨA

2 - 2ΨA

ΨB

+ ΨB

2 .

Molecular Orbital Theory (5)

The two orbitals Ψb

and Ψa

differ from

each other.

In the bonding molecular orbital, the

wave functions for the component atoms

reinforce each other in the region

between the nuclei.

Ψb2 is the electron distribution in the

hydrogen molecule, also known as the

probability function.

ΨA

and ΨB

Ψb = Ψ

A+ Ψ

B

Ψb

2

Molecular Orbital Theory (6)

Note that in the bonding situation, the

electrons are in between two nuclei.

In the anti-bonding orbital , the wave

functions cancel, forming a node between

the nuclei.

In this situation, the electrons are not in

between the two nuclei.

ΨA

and ΨB

Ψa = Ψ

A- Ψ

B

Ψa

2

Correlation Energy Diagrams (1)

The diagram in the opposite panel shows

the energy correlation diagramenergy correlation diagram of the

molecular orbitals and the atomic

orbitals.

The atomic orbitals are both on the

right and the left of the diagram.

The energy levels in the centre are those

of the corresponding bonding (1sσ) and

anti-bonding (1sσ*

) molecular orbitals.

1sA 1s

B

1sσ

1sσ*

↑ ↑

Energy

Notice that the bonding orbitals have a lower energy level compared to the atomic orbitals - one of the reasons for bonding is to achieve lower energy levels

Correlation Energy Diagrams (2)

In H2, each atom provides 1 electron to

the molecular orbital.

This results in the two electrons pairing

up in the 1sσ

bonding orbital (Hund’s

rule, Pauli Exclusion principle).

This results in a stable bonding between

H-H atoms, and this is exemplified

experimentally.

1sA

1sB

1sσ

1sσ*

↑ ↑

Energy

↑↓

Correlation Energy Diagrams (3)

How about H2+?

In this case, the dihydrogen cation has

only one electron and therefore has only

one electron in the bonding molecular

orbital of 1sσ.

Although only one electron exists, this

provides an energy stabilisation of ∆E

compared to the atomic orbitals.

In the case of H2, it has a total energy

stabilisation of 2∆E.

1sA 1s

B

1sσ

1sσ*

↑

Energy

↑∆E

Correlation Energy Diagrams (4)

How about H2-?

For the dihydrogen anion, there are three electrons to distribute to the molecular orbitals, and the third electron populates the 1sσ* orbital.

This is an anti-bonding orbital and will result in the electron gaining destabilisation energy of ∆E’ .

Assuming ∆E = ∆E’ , the bond energy for H2

- is then 2∆E – ∆E’ = ∆E (similar to that of H2

+).

1sA 1s

B

1sσ

1sσ*

↑↓ ↑

Energy

↑↓

↑

∆E

∆E’

Energy Correlation Diagrams (5)

How about He2?

In He2, we have a total of four electrons to populate into the molecular orbitals.

The third and fourth electron goes into the 1sσ* orbital, and since it is an anti-bonding orbital, the net stabilisation obtained from the bonding orbital is lost as it is cancelled by the anti-bonding orbitals.

As a result, He2 is predicted to be unstable.

1sA 1s

B

1sσ

1sσ*

↑↓ ↑↓

Energy

↑↓

↑↓

Considerations for Molecular Orbital formation

In order to determine which atomic orbitals interact to form molecular

orbitals, we have to consider the factors that impact on the interaction

between atomic orbitals.

They are:

* The energy differencebetween the interacting orbitals must be small,

* The overlap between the orbitals must be large.

This means that the 2s orbitals will only interact with other 2s orbitals

and not with the 1s or 2p orbitals.

Molecules with Electrons in the 2s Orbitals (1)

When we include the 2s atomic orbitals

into the formation of molecular orbitals, we

combine them in the same fashion as we

did for the 1s atomic orbitals.

The 2sσ

results from 2sA

+ 2sB, while 2s

σ*

results from 2sA

- 2sB.

The molecular orbital diagram appears as

in the panel.

2sA 2s

B

2sσ

2sσ*

1sA 1s

B

1sσ

1sσ*

Molecules with Electrons in the 2s Orbitals (2)

In the case of Li2, there are a total of 3

electrons for an atomic configuration of

1s2 2s1.

These will combine as shown in the panel

- only a single bond results.

Note that the 1s orbitals do not form

bonds (as we expect) from the MO theory

as it is cancelled out by the anti-bonding

orbitals.

The electronic configuration of Li2

is then

(1sσ)2 (1s

σ*)2 (2s

σ)2.

2sA 2s

B

2sσ

2sσ*

1sA 1s

B

1sσ

1sσ*

↑

↑↓ ↑↓

↑↓

↑↓

↑↓

↑

Molecules with Electrons in the 2s Orbitals (3)

For Be2, there are four electrons for each

Be atom thus giving an electronic configuration of 1s2 2s2.

The MO for Be2

is as shown in the opposite panel.

The electrons populating the 2s molecular orbitals gives use an electronic configuration of(1s

σ)2(1s

σ*)2(2s

σ)2 (2s

σ*)2.

Be2

is not expected to be stable, and thus will not exist.

2sA 2s

B

2sσ

2sσ*

1sA 1s

B

1sσ

1sσ*

↑↓

↑↓ ↑↓

↑↓

↑↓

↑↓

↑↓

↑↓

Molecules with Electrons in the 2p Orbitals (1)

There are three 2p orbitals, each of these orbitals are in the x, y and z directions of the Cartesian coordinates.

Elements such as B, C, N, O, F, ... have electronic configurations that utilise the p-orbitals.

Here, if the p-orbital is lying along the internuclear axis, then it can undergo symmetric combination to give a bonding orbital.

Alternatively, it can undergo anti-symmetric combination to give an anti-bonding orbital .

Symmetric

Anti-symmetric

Molecules with Electrons in the 2p Orbitals (2)

The σ2p

molecular orbital has bulk of the electron density between the nuclei, thus contributes towards the bonding.

The σ*2p

molecular orbital has the electron density mainly outwith the internuclear area, thus does not contribute towards the bonding.

Molecules with Electrons in the 2p Orbitals (3)

The remaining p-orbitals (px and py) do not overlap, but they lie

perpendicular to the internuclear axis.

They can undergo either plus or minus combination to give 2pπ orbitals

and are called doubly degenerate as the orbitals formed are equal in

energy and size but differ in direction.

MO Theory in Bonding of O2 (1)

O has eight electrons; O2

has 16 electrons.It has the following electronic configuration

(1sσ)2 (1s

σ*)2 (2s

σ)2 (2s

σ*)2 (2p

σ)2 (2p

π)4 (2p

π*)2

There are 4 nett bonding electrons.

2pσ

2pσ*

1s 1s

1sσ

1sσ*

↑↓

↑↓↑↓

↑↓

2s 2s

2sσ

2sσ*

↑↓

↑↓↑↓

↑↓

↑↓ 2pπ

2pπ*

2p

↑↓↑↓

↑ ↑

↑↓ ↑↑ ↑↓ ↑↑2p

MO Theory in Bonding of O2

(2)

4 (2pπ) + 2 (2p

σ) - 2 (2p

π*)

Hence there are 4/2 = 2 bondspresent (O=O). There are also 2 anti-bonding unpaired electrons.

This explains the paramagnetic behavior of dioxygen.

2pσ

2pσ*

1s 1s

1sσ

1sσ*↑↓

↑↓↑↓

↑↓

2s 2s

2sσ

2sσ*

↑↓

↑↓↑↓

↑↓

↑↓ 2pπ

2pπ*

2p↑↓↑↓

↑ ↑↑↓ ↑↑ ↑↓ ↑↑

2p

MO Theory in Bonding of F2

(1)F has 9 electrons while F

2has 18 electrons.

It has the following electronic configuration

(1sσ)2 (1s

σ*)2 (2s

σ)2 (2s

σ*)2 (2p

σ)2 (2p

π)4 (2p

π*)4

There are 2 nett bonding electrons and a single bond(F-F).

There are four anti-bonding electrons, thus F2

is highly reactive species.

2pσ

2pσ*

1s 1s

1sσ

1sσ*↑↓

↑↓↑↓

↑↓

2s 2s

2sσ

2sσ*↑↓

↑↓↑↓

↑↓

↑↓ 2pπ

2pπ*

2p↑↓

↑↓

↑↓ ↑ ↑↓ ↑

2p

↑↓ ↑↓

↑↓↑↓

2sσ*

↑↓

MO Theory in Bonding of N2

N2

has a different energy level compared to those of O2

and F2.

N2

has the following electronic structure

(1sσ)2 (1s

σ*)2 (2s

σ)2 (2s

σ*)2 (2p

π)4 (2p

σ)2

There are 6 nett bonding electrons, giving 3 bonds.

N2

is therefore very stable and unreactive as there are no anti-bonding electrons.

2pσ

2pσ*

1s 1s

1sσ

1sσ*↑↓

↑↓↑↓

↑↓

2s 2s

2sσ

↑↓

↑↓↑↓

↑↓

↑↓2p

π

2pπ*

2p↑↓

↑ ↑2p

↑↑↑↑

MO Theory in Bonding of Ne2

Ne has 10 electrons and Ne2 has 20 electrons.It has the following electronic structure

(1sσ)2 (1s

σ*)2 (2s

σ)2 (2s

σ*)2 (2p

σ)2 (2p

π)4 (2p

π*)4 (2p

σ*)2

This means that there are no nett bonding electrons and Ne2 will not exist.

2pσ

2pσ*

1s 1s

1sσ

1sσ*

↑↓

↑↓↑↓

↑↓

2s 2s

2sσ

2sσ*

↑↓

↑↓↑↓

↑↓

↑↓ 2pπ

2pπ*

2p

↑↓↑↓

↑↓ ↑↓

2p

↑↓ ↑↓

↑↓↑↓

↑↓ ↑↓

↑↓

Topics Covered in this Lecture

Concepts of Molecular OrbitalsCorrelation Energy DiagramPrediction of Reactivity of Molecules