Biophysical ChemistryChemistry 24a Winter Term 2009-10Instructor: Sunney I. ChanLecture 5 January 13, 2010Atomic Structure and Chemical BondingThe Hydrogen AtomThe HamiltonianThe Hamiltonian (Ĥ) is the sum of the kinetic energies and potential energies associated with an atomic and molecular system:Ĥisolated molecule(p1, r1; p2, r2;., pi, ri;..)= Σipi2/2mi+ V(r1; r2;.. ri;…)kinetic energy potential energywhere pi, riand mirefer to the momentum, coordinates, and the mass of the ithparticle (electrons and nuclei), respectively.Typically for a moleculeV(r1; r2;.. ri;…)= ΣNΣk(-zNe2/rNk) (coulomb attraction between nuclei and electrons)+ Σk’≠k (e2/rkk’) (electrostatic repulsion between electrons)+ ΣN’≠N zNzN’e2/RNN’(nuclear-nuclear repulsion)whereΣkdenotes sum over all electrons;and ΣNdenotes sum over all nuclei in the molecule.
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For an atomV(r1; r2;.. ri;…)= Σk(-zNe2/rNk) (coulomb attraction between nucleus and electrons)+ Σk’≠k (e2/rkk’) (electrostatic repulsion between electrons)whereΣkdenotes sum over all the electrons in the atom.For the hydrogen atomV(r) = - e2/r(coulomb attraction between proton and electron) So, Ĥatom= -ħ2/2μ▽2-e2/r(when kinetic energy is expressed relative to the center of mass of the two-particle system)μ= Mpme/(Mp+ me) ≈me (since Mp>> me)(reduced mass)where Mpand meare the masses of the proton and electron, respectively. For the hydrogen atom,Schrőedinger equation for the problem:ĤatomΦn(r,θ,φ) = - (ħ2/2me)▽2Φn(r,θ,φ) - (e2/r) Φn(r,θ,φ) = EnΦn(r,θ,φ) where ris the magnitude of the vector r, namely, just the distance of the electron from the proton.Φnis the wavefunction associated with the energy En.Polar coordinatesrrxyzθφz = rcosθx = rsin θcosφy = rsin θsin φProblem has spherical symmetry!