Chemistry 445.

Lecture 3.

Molecular Orbital Theory

We start by reminding ourselves of the shapes and signs of the wavefunction on the atomic orbitals. Below are the s and three p orbitals, showing boundary surfaces (H&S Fig. 1.9)

Note: Pink color indicates sign of wavefunction opposite to that ofthe white part of the orbital.

The five atomic d-orbitals

Orbitals and Quantum numbers:The solution to the Schrödinger wave-

equation leads to a set of wavefunctions that yields 4 types of quantum numbers instead of the single quantum number yielded by the Bohr model.

These are:

1) The principal quantum number, n, which has values of 1,2,3,… This corresponds to the quantum number n in the Bohr model of the atom.

Quantum numbers (contd.)

2) The Azimuthal quantum number l, which has values of 0 to (n-1) for each value of n. The different values of l correspond to orbital types as follows:

l = 0 1 2 3letter used = s p d f

3) The magnetic quantum number ml, can have values of –l through 0 to +l for each value of l.

Value of l possible values of ml

0 01 -1,0,+12 -2,-1,0,+1,+23 -3,-2,-1,0,+1,+2,+3

4) The spin quantum number (ms). This can have values of +½ or –½. This means that for each value of ml there are two values of ms. It is this that leads to the occupation of each orbital by two electrons of opposite spin, i.e. with ms = +½ or –½.

These quantum numbers lead to the shells (different values of n) and subshells (different values of l) that lead to our modern understanding of chemistry. The number of orbitals in each sub-shell (1 for s, 3 for p, 5 for d, and 7 for f sub-shells) is determined by ml, and ms determines that only two electrons of opposite spin can occupy each orbital.

Schrödinger’s model:

z

x

y

s-orbital p-orbital(1 of 3)

d-orbital(1 of 5)

f-orbital(1 of 7)

Representation of orbitals:

The H-atom compared to many-electron atoms:

ener

gy

1s

2s 2p

3s 3p 3d

n = 1

n = 3

n = 2

ener

gy

1s

2s

3s

n = 1

n = 3

n = 2

3d

3p

2p

4p

4s

H-atom. All subshells within Many-electron atom. Subshellssame shell have the same energy. within same shell have different energies.

n = 4n = 44s 4p 4d 4f

The essence of MO theory is that overlap of two orbitals always occurs in two ways. In one (bottom), the two 1s orbitals shown here overlapping have the same sign of the wavefunction, and so a net overlap occurs. This produces a lower energy bonding orbital. In the upper case, the two orbitals are of opposite sign, and so no net overlap occurs. This produces a higher energy anti-bonding orbital.

+

+

sign of wavefunction is the same

Sign of wavefunction is opposite higher energyanti-bonding orbital

lower energybonding orbital

1s 1s

1s 1s

σ*1s

σ1s

Drawing up a Molecular Orbital (MO) diagram for H2

1s atomicorbital of H atom

1s atomicorbital of H atom

energy arrow representselectron in 1s orbital

energy levelof 1s orbitalof H-atom

Drawing up a Molecular Orbital (MO) diagram for H2

1s atomicorbital of H atom

1s atomicorbital of H atom

energy

These are the molecularorbitals of the H2 molecule

σ*1s anti-bondingmolecularorbital in H2

molecule

σ1s bondingmolecularorbital in H2

molecule

Molecular Orbital (MO) diagram for H2 molecule (bond order = 1)

1s atomicorbital of H

1s atomicorbital of H

asterisk denotesanti-bonding orbital

σ* 1s

σ 1s

arrow = electron

atomatom

Some observations on MO diagrams:

the two arrows areopposite in directionindicating a pair of spin-paired electrons of opposite spin

because of the Pauli exclusion Principle each orbital can containa maximum of two electrons, which must be of opposite spin

in labeling the molecularorbitals, the type of overlap is specified (σ or π), and the atomic orbitals involved indicated.

A single bond consists of a shared pair ofelectrons (Lewis). InMO theory BondOrder (BO) = (No. of e’s inbonding levels –no. of e’s in anti-bonding levels)/2

BO for H2 = (2-0)/2 = 1

Some more observations on MO diagrams:

The greater thedrop in energythe stronger thebond. For the H2

molecule thedrop is 218 kJ/molso the enthalpy ofdissociation of theH2 molecule is436 kJ/mole

In MO theory the reason molecules form isbecause the bonding orbitals formed are lowerin energy than the atomic orbitals, and theelectrons are lowered in energy by this amount.

Even more observations on MO diagrams:

Photon ofEnergy = hv

Electron excitedto anti-bonding level

MO diagrams show how a photon of energy = hv = the difference in energybetween two MO’s, can cause an electron to be excited to the higher energy level MO. In this excited state the bond order = zero and so the H2 molecule can photo-dissociate. Whether the transition can occur is also determined by the parity of the orbitals (g or u) – see later.

BO = (1-1)/2 = 0for excited state

Identification of bonding and non-bonding molecular orbitals.

A bonding MO has no nodal plane between the two atoms forming the bond, i.e. the electron density does not go to zero at a node. An anti-bonding MO has a nodal plane where electron-density = zero:

nodal plane

σ(1s) bonding orbital σ*(1s) anti-bonding orbital

π(2p) bonding orbital π*(2p) anti-bonding orbital