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Unformatted text preview: chem 260/261 Lecture 07 9/21/2011 1 Chem 260/261 Lecture 7 21/9/2011 Wed, September 21, 2011 Today in Chemistry 260/261 • Finish H hydrogen atom orbitals and energies. •Multi-electron atoms • Aufbau exceptions/oddities • Periodic trends Next in Chemistry 260/261 • reading: 5.5 • Aufbau exceptions/oddities • Periodic trends • many-electron atoms, periodic properties Wed, September 21, 2011 Outline • Hydrogen Atom electronic states: Energies and Wavefunctions • Interpreting the wave function. • Understanding Hydrogen Orbitals: Radial distributions: s function (l=0) p,d,f function (l=1,2,3) • Yet another quantum number. The spin. Remember each orbital is a possible solution for the hydrogen atom system. Wed, September 21, 2011 Schrödinger's equation • QM teaches us to assign wave-particle duality properties • The wave function is the mathematical tool to describe this duality • Schrodinger eq . has its relation/origins to wave mechanics (discretization and standing wave frequencies) • à the second derivative in space (curvature) is also the kinetic energy (change in time) • V is the rest of the energy terms due to forces acting on the electrons • Together they deDine the Hamiltonian • The Hamiltonian acting on the wave function gives you back the energy of the system . ! ! 2 2m e d 2 " dx 2 + d 2 " dy 2 + d 2 " dz 2 # $ % & ' ( + ˆ V x,y,z ( ) " = E " Wed, September 21, 2011 Schrödinger's equation Wed, September 21, 2011 Applying Schrödinger's equation to the hydrogen atom p + e- r z y x Schrödinger's equation for an electron in a 3¡ box with a proton: Coulomb ` s Law Therefore, it is natural to describe the atom using SPHERICAL (POLAR) COOR¡INATES But the H atom has a spherically symmetric potential (assuming the proton remains stationary)… it would be very inconvenient to describe the electron as being in a l box z Ù V r ( ) = − Ze 2 4 πε r ! ! 2 2m e d 2 " dx 2 + d 2 " dy 2 + d 2 " dz 2 # $ % & ' ( + ˆ V x,y,z ( ) " = E " Wed, September 21, 2011 chem 260/261 Lecture 07 9/21/2011 2 H atom wave function summary ψ ( r , θ , φ ) = R ( r ) Θ ( θ ) Φ ( φ ) Solutions depend on three numbers that are related – they correspond to different possible states of an electron in H atom n = 1,2,3,4... l = 0,1,2,3...,( n − 1) m = − l ,( − l + 1),...,0,...( l − 1), l ψ nlm r , θ , φ ( ) = R nl r ( ) Y lm θ , φ ( ) R nl r ( ) = N nl L nl e − r na • Energy depends only on n. • Node is where the distribution changes sign (as n increases the number of nodes increases) • The probability at any point to Fnd electron is always zero • Can discuss non-vanishing probabilities at REGIONs ( l cloud z ) Wed, September 21, 2011 The Hydrogen Atom The Schrödinger equation for our l electron in a sphere with a proton z model is now completely solved....
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This note was uploaded on 01/19/2012 for the course CHEM 260 taught by Professor Staff during the Fall '08 term at University of Michigan.
- Fall '08