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Chemistry 2 Lecture 1 Quantum Mechanics in Chemistry Your lecturers 8am 12pm Assoc. Prof Timothy Schmidt Assoc. Prof. Adam J Bridgeman Room 315 Room 222 firstname.lastname@example.org email@example.com 93512781 93512731 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 valued Must be continuous Should be differentiable (where potential is) The Schrödinger equation • Observable energy levels of a quantum particle obey this equation wavefunction Hamiltonian operator Energy Hydrogen atom • You already know the solutions to the Schrödinger equation! • For hydrogen, they are orbitals… 3s 2s 1s 2p 3p No nodes 1 node 2 nodes Hydrogen energy levels E 2 nodes 3p (also 3d orbitals…) 3s 2s 2p 1 node Rydberg constant energy level Z 2 En 2 1s n Nuclear charge No nodes Principal quantum number ionization potential 4s 4p 3s 3p 2s 2p UV (can’t see) 1s Solutions to Schrödinger equation • Hamiltonian operates on wavefunction and gets function back, multiplied by energy. ˆ H • 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 ˆ H 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 H2+ Revision – H2+ • Near each nucleus, electron should behave as a 1s electron. • At dissociation, 1s orbital will be exact solution at each nucleus r Revision – H2+ • At equilibrium, we have to make the lowest energy possible using the 1s functions available r ? r Revision – H2+ = 1sA – 1sB anti-bonding 1sA 1sB E 1sA 1sB = 1sA + 1sB bonding 1sA 1sB Revision – H2 = 1sA – 1sB anti-bonding 1sA 1sB E 1sA 1sB = 1sA + 1sB bonding 1sA 1sB Revision – He2 = 1sA – 1sB anti-bonding 1sA 1sB E 1sA 1sB = 1sA + 1sB bonding 1sA 1sB NOT BOUND!! 2nd row homonuclear diatomics • Now what do we do? So many orbitals! 2p 2p 2s 2s 1s 1s 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 total energy operator (Hamiltonian) ˆ 2 H 1d integral over all space orbital wavefunctions Interacting orbitals 1. Orbitals interact inversely proportionally to their energy difference. Orbitals of the same energy interact completely, yielding completely mixed linear combinations. 1 ( 2 sA 2 sB ) 2 2p 2p 2s 2s 2 sA 1 2 ( 2 sA 2 sB ) 2 sB 1s 1s Interacting orbitals 1. The extent of orbital mixing is given by the integral ˆ 2 H 1d something 2p 2p 2s 2 s 2 p 2s 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. 1s 1s Interacting orbitals 1. The extent of orbital mixing is given by the integral ˆ 2 H 1d 0 cancels 2p 2p 2s 2 s 2 p 2s There is no net interaction between these orbitals. The positive-positive term is cancelled by the positive-negative term 1s 1s (First year) MO diagram Orbitals interact most with the corresponding orbital on the other atom to make perfectly mixed linear combinations. (we ignore core). 2p 2p 2s 2s 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 2 p * 2 p * 2 p 2 p 2s * diamagnetic N2 Bond Order = 3 2s More refined MO diagram orbitals can now interact 2 p * 2 p * 2 p 2 p 2s * 2s More refined MO diagram * orbitals can interact * 2 p * 2 p * 2 p 2 p 2s * * 2s More refined MO diagram orbitals do not interact * 2 p * 2 p * * 2 p 2 p 2s * * 2s More refined MO diagram sp mixing * 2 p * 2 p * * 2 p 2 p This new interaction energy 2s * * Depends on the energy spacing between the 2s and the 2p 2s Smallest energy gap, sp mixing Largest energy gap, and thus smallest mixing and thus largest between 2s and 2p is for mixing between 2s Fluorine. and 2p is for Boron. 2p Z 2 2s c.f. En 2 n sp mixing * * * * * * * * * * * * weakly bound paramagnetic diamagnetic Be2 B2 C2 N2 sp mixing in N2 sp mixing in N2 -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 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 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.
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