Chemistry 2 Lecture 1 Quantum Mechanics in Chemistry

Your lecturers 12pm Assoc. Prof. Adam J Bridgeman Room am Dr Timothy Schmidt Room

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:  x  p > 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 valued Must be continuous Should 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 3p No nodes 1 node 2 nodes

Hydrogen energy levels 1s 2s 2p 3s No nodes 1 node 2 nodes 3p 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.

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.

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 H 2 +

Revision – H 2 + Near each nucleus, electron should behave as a 1s electron. At dissociation, 1s orbital will be exact solution at each nucleus  r

Revision – H 2 + At equilibrium, we have to make the lowest energy possible using the 1s functions available  r  r ?

Revision – H 2 +  1s A 1s B  1s A 1s B  = 1s A – 1s B 1s A 1s B E bonding anti-bonding  = 1s A + 1s B

Revision – H 2  1s A 1s B  1s A 1s B  = 1s A – 1s B 1s A 1s B E bonding anti-bonding  = 1s A + 1s B

Revision – He 2  1s A 1s B  1s A 1s B  = 1s A – 1s B 1s A 1s B E bonding anti-bonding  = 1s A + 1s B NOT BOUND!!

2 nd row homonuclear diatomics Now what do we do? So many orbitals! 1s 2s 2p

Interacting orbitals Orbitals 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 orbital wavefunctions integral over all space total energy operator (Hamiltonian)

Interacting orbitals 1.Orbitals interact inversely proportionally to their energy difference. Orbitals of the same energy interact completely, yielding completely mixed linear combinations. 1s 2s 2p

Interacting orbitals 1.The extent of orbital mixing is given by the integral 1s 2s 2p 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 orbitals 1.The extent of orbital mixing is given by the integral 1s 2s 2p cancels There is no net interaction between these orbitals. The positive-positive term is cancelled by the positive-negative term

(First year) MO diagram Orbitals interact most with the corresponding orbital on the other atom to make perfectly mixed linear combinations. (we ignore core). 2s 2p

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

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.  *. 

Molecular Orbital Theory - Revision Can predict bond strengths qualitatively N 2 Bond Order = 3 diamagnetic

More refined MO diagram  orbitals can now interact

More refined MO diagram  orbitals can interact

More refined MO diagram  orbitals do not interact

More refined MO diagram sp mixing This new interaction energy Depends on the energy spacing between the 2s  and the 2p 

sp mixing 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 Be 2 B2B2 C2C2 N2N2 weakly boundparamagnetic diamagnetic

sp mixing in N 2

 -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 H 2 +. 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 Questions 1.Which of the wavefunctions (a) – (d) is acceptable as a solution to the Schrödinger equation? 2.Why is s-p mixing more important in Li 2 than in F 2 ? 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 N 2, NO, CO and O 2 are 15.6, 9.25, 14.0 and 12.1 eV respectively. Explain.