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 of the 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 =0123 letter used= spdf 3) The magnetic quantum number m l, can have values of – l through 0 to + l for each value of l. Value of l possible values of m l ,0, ,-1,0,+1, ,-2,-1,0,+1,+2,+3

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

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 energy anti-bonding orbital lower energy bonding orbital 1s1s1s1s 1s1s1s1s σ*1s σ1sσ1s

Drawing up a Molecular Orbital (MO) diagram for H 2 1s atomic orbital of H atom 1s atomic orbital of H atom energy arrow represents electron in 1s orbital energy level of 1s orbital of H-atom

Drawing up a Molecular Orbital (MO) diagram for H 2 1s atomic orbital of H atom 1s atomic orbital of H atom energy These are the molecular orbitals of the H 2 molecule σ*1s anti-bonding molecular orbital in H 2 molecule σ1s bonding molecular orbital in H 2 molecule

Molecular Orbital (MO) diagram for H 2 molecule (bond order = 1) 1s atomic orbital of H 1s atomic orbital of H asterisk denotes anti-bonding orbital σ* 1s σ 1s arrow = electron atom

Some observations on MO diagrams: the two arrows are opposite in direction indicating a pair of spin-paired electrons of opposite spin because of the Pauli exclusion Principle each orbital can contain a maximum of two electrons, which must be of opposite spin in labeling the molecular orbitals, the type of overlap is specified (σ or π), and the atomic orbitals involved indicated. A single bond consists of a shared pair of electrons (Lewis). In MO theory Bond Order (BO) = (No. of e’s in bonding levels – no. of e’s in anti- bonding levels)/2 BO for H 2 = (2-0)/2 = 1

Some more observations on MO diagrams: The greater the drop in energy the stronger the bond. For the H 2 molecule the drop is 218 kJ/mol so the enthalpy of dissociation of the H 2 molecule is 436 kJ/mole In MO theory the reason molecules form is because the bonding orbitals formed are lower in energy than the atomic orbitals, and the electrons are lowered in energy by this amount.

Even more observations on MO diagrams: Photon of Energy = hv Electron excited to anti-bonding level MO diagrams show how a photon of energy = hv = the difference in energy between 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 H 2 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 = 0 for 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