Chemistry 2 Lecture 1 Quantum Mechanics in Chemistry.

Chemistry 2

Lecture 1 Quantum Mechanics in Chemistry

Your lecturers

12pmAssoc. Prof. Adam J Bridgeman

Room 222 [email protected]

93512731

8amDr Timothy Schmidt

Room 457 [email protected]

93512781

Learning outcomes

• Be able to recognize a valid wavefunction in terms of its being single valued, continuous, and differentiable (where potential is).

• Be able to recognize the Schrödinger equation. Recognize that atomic orbitals are solutions to this equation which are exact for Hydrogen.

• Apply the knowledge that solutions to the Schrödinger equation are the “observable energy levels” of a molecule.

• Use the principle that the mixing between orbitals depends on the energy difference, and the resonance integral.

• Rationalize differences in orbital energy levels of diatomic molecules in terms of s-p mixing.

How quantum particles behave• Uncertainty principle: DxDp > ℏ• Wave-particle duality

• Need a representation for a quantum particle: the wavefunction (ψ)

How quantum particles behave

The Wavefunction

• The quantum particle is described by a function – the wavefunction – which may be interpreted as a indicative of probability density

• P(x) = Ψ(x)2

• All the information about the particle is contained in the wavefunction

The Wavefunction

Must be single valuedMust be continuousShould be differentiable (where potential is)

The Schrödinger equation

• Observable energy levels of a quantum particle obey this equation

Energy

wavefunction

Hamiltonian operator

Hydrogen atom

• You already know the solutions to the Schrödinger equation!

• For hydrogen, they are orbitals…

1s

2s

2p

3s

3pNo nodes

1 node 2 nodes

Hydrogen energy levels

1s

2s 2p

3s

No nodes

1 node

2 nodes

3p

2

2

n

ZEn

energy level

Rydberg constant

Principal quantum number

Nuclear charge

(also 3d orbitals…)

E

1s

2s 2p

3s 3p

4s 4p

ionization potential

UV (can’t see)

Solutions to Schrödinger equation

• Hamiltonian operates on wavefunction and gets function back, multiplied by energy.

• More nodes, more energy (1s, 2s &c..)• Solutions are observable energy levels.• Lowest energy solution is ground state.• Others are excited states.

H

Solutions to Schrödinger equation

Exactly soluble for various model problems• You know the hydrogen atom (orbitals)

In these lectures we will deal with • Particle-in-a-box• Harmonic Oscillator• Particle-on-a-ring

But first, let’s look at approximate solutions.

H

Approximate solutions to Schrödinger equation

• For molecules, we solve the problem approximately.

• We use known solutions to the Schrödinger equation to guide our construction of approximate solutions.

• The simplest problem to solve is H2+

Revision – H2+

• Near each nucleus, electron should behave as a 1s electron.

• At dissociation, 1s orbital will be exact solution at each nucleus

Y

r

Revision – H2+

• At equilibrium, we have to make the lowest energy possible using the 1s functions available

Y

r

Y

r ?

Revision – H2+

Y

1sA 1sB

Y

1sA 1sB

Y = 1sA – 1sB

1sA1sB

E

bonding

anti-bonding

Y = 1sA + 1sB

Revision – H2

Y

1sA 1sB

Y

1sA 1sB

Y = 1sA – 1sB

1sA1sB

E

bonding

anti-bonding

Y = 1sA + 1sB

Revision – He2

Y

1sA 1sB

Y

1sA 1sB

Y = 1sA – 1sB

1sA1sB

E

bonding

anti-bonding

Y = 1sA + 1sB

NOT BOUND!!

2nd row homonuclear diatomics

• Now what do we do? So many orbitals!

1s 1s

2s 2s

2p2p

Interacting orbitalsOrbitals can interact and combine to make new approximate solutions to the Schrödinger equation. There are two considerations:1. Orbitals interact inversely proportionally to their energy

difference. Orbitals of the same energy interact completely, yielding completely mixed linear combinations.

2. The extent of orbital mixing is given by the integral

dH 12ˆ

orbital wavefunctions

integral over all space

total energy operator (Hamiltonian)

Interacting orbitals1. Orbitals interact inversely proportionally to their energy

difference. Orbitals of the same energy interact completely, yielding completely mixed linear combinations.

1s 1s

2s 2s

2p2p

sBsA 222

1

sBsA 222

1

sA2 sB2

Interacting orbitals1. The extent of orbital mixing is given by the integral

somethingˆ12 dH

1s 1s

2s 2s

2p2p

s2 p2

The 2s orbital on one atom can interact with the 2p from the other atom, but since they have different energies this is a smaller interaction than the 2s-2s interaction. We will deal

with this later.

Interacting orbitals1. The extent of orbital mixing is given by the integral

0ˆ12 dH

1s 1s

2s 2s

2p2p

s2 p2

cancels

There is no net interaction between these orbitals.The positive-positive term is cancelled by the positive-negative term

(First year) MO diagramOrbitals interact most with the corresponding orbital on the other atom

to make perfectly mixed linear combinations. (we ignore core).

2s 2s

2p2p

Molecular Orbital Theory - Revision

• Molecular orbitals may be classified according to their symmetry• Looking end-on at a diatomic molecule, a molecular orbital may

resemble an s-orbital, or a p-orbital.• Those without a node in the plane containing both nuclei resemble an

s-orbital and are denoted s-orbitals.• Those with a node in the plane containing both nuclei resemble a p-

orbital and are denoted p-orbitals.

s p

Molecular Orbital Theory - Revision

• Molecular orbitals may be classified according to their contribution to bonding

• Those without a node between the nuclei resemble are bonding.• Those with a node between the nuclei resemble are anti-bonding,

denoted with an asterisk, e.g. s*.

*s *p

Molecular Orbital Theory - Revision• Can predict bond strengths qualitatively

N2 Bond Order = 3

*2 p

*2 p

p2

p2

s2

*2 s

diamagnetic

More refined MO diagrams orbitals can now interact

*2 p*2 p

p2

p2

s2

*2 s

More refined MO diagram*s orbitals can interact

*2 p*2 p

p2

p2

s2

*2 s

*

*

More refined MO diagramp orbitals do not interact

*2 p*2 p

p2

p2

s2

*2 s

*

*

*

More refined MO diagramsp mixing

*2 p*2 p

p2

p2

s2

*2 s

*

*

*

This new interaction energyDepends on the energy spacing between the 2ss and the 2ps

sp mixing

2

2

n

ZEn

c.f.

2p

2s

Smallest energy gap, and thus largest mixing between 2s and 2p is for Boron.

Largest energy gap, and thus smallest mixing between 2s and 2p is for Fluorine.

sp mixing

*

*

*

Be2

*

*

*

B2

*

*

*

C2

*

*

*

N2

weakly bound paramagnetic diamagnetic

sp mixing in N2

sp mixing in N2

s-bonding orbital Linear combinations of lone pairs

:N≡N:

Summary• A valid wavefunction is single valued, continuous, and

differentiable (where potential is).• Solving the Schrödinger equation gives the observable energy

levels and the wavefunctions describing a sysyem:• Atomic orbitals are solutions to the Schrödinger equation for

atoms and are exact for hydrogen.• Molecular orbitals are solutions to the Schrödinger equation for

molecules and are exact for H2+.

• Molecular orbitals are made by combining atomic orbitals and are labelled by their symmetry with σ and π used for diatomics

• Mixing between atomic orbitals depends on the energy difference, and the resonance integral.

• The orbital energy levels of diatomic molecules are affected by the extent of s-p mixing.

Next lecture

• Particle in a box approximation– solving the Schrödinger equation.

Week 10 tutorials

• Particle in a box approximation– you solve the Schrödinger equation.

Practice Questions1. Which of the wavefunctions (a) – (d) is

acceptable as a solution to the Schrödinger equation?

2. Why is s-p mixing more important in Li2 than in F2?

3. The ionization energy of NO is 9.25 eV and corresponds to removal of an antibonding electron

(a) Why does ionization of an antibonding electron require energy?

(b) Predict the effect of ionization on the bond length and vibrational frequency of NO

(c) The ionization energies of N2, NO, CO and O2 are 15.6, 9.25, 14.0 and 12.1 eV respectively. Explain.