L5 - Biophysical Chemistry Chemistry 24a Winter Term...

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Biophysical Chemistry Chemistry 24a Winter Term 2009-10 Instructor: Sunney I. Chan Lecture 5 January 13, 2010 Atomic Structure and Chemical Bonding The Hydrogen Atom The Hamiltonian The Hamiltonian ( Ĥ ) is the sum of the kinetic energies and potential energies associated with an atomic and molecular system: Ĥ isolated molecule (p 1 , r 1 ;p 2 , r 2 ; ., p i , r i ; .. ) = Σ i p i 2 /2m i + V(r 1 ; r 2 ;.. r i ;…) kinetic energy potential energy where p i , r i and m i refer to the momentum, coordinates, and the mass of the i th particle (electrons and nuclei), respectively. Typically for a molecule V(r 1 ; r 2 ;.. r i ;…) = Σ N Σ k (-z N e 2 /r Nk ) (coulomb attraction between nuclei and electrons) + Σ k’ k (e 2 /r kk’ ) (electrostatic repulsion between electrons) + Σ N’ N z N z N’ e 2 /R NN’ (nuclear-nuclear repulsion) where Σ k denotes sum over all electrons; and Σ N denotes sum over all nuclei in the molecule.
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For an atom V(r 1 ; r 2 ;.. r i ;…) = Σ k (-z N e 2 /r Nk ) (coulomb attraction between nucleus and electrons) + Σ k’ k (e 2 /r kk’ ) (electrostatic repulsion between electrons) where Σ k denotes sum over all the electrons in the atom. For the hydrogen atom V(r) = - e 2 / r (coulomb attraction between proton and electron) So, Ĥ atom = - ħ 2 /2 μ 2 -e 2 / r (when kinetic energy is expressed relative to the center of mass of the two-particle system) μ = M p m e /(M p + m e ) m e (since M p >> m e ) (reduced mass ) where M p and m e are the masses of the proton and electron, respectively. For the hydrogen atom, Schr ő edinger equation for the problem: Ĥ atom Φ n ( r , θ , φ ) = - ( ħ 2 /2m e ) 2 Φ n ( r , θ , φ ) -(e 2 / r ) Φ n ( r , θ , φ ) = E n Φ n ( r , θ , φ ) where r is the magnitude of the vector r, namely, just the distance of the electron from the proton. Φ n is the wavefunction associated with the energy E n . Polar coordinates r r x y z θ φ z = r cos θ x = r sin θ cos φ y = r sin θ sin φ Problem has spherical symmetry!
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In cartesian coordinates: 2 Φ n = 2 Φ n / x 2 + 2 Φ n / y 2 + 2 Φ n / z 2 In polar coordinates: 2 Φ n = (1/r 2 ) ( / r)(r 2 Φ n / r) + (1/r 2 )(1/sin θ )( / θ )(sin θ Φ n / θ )+ (1/r 2 )(1/sin 2 θ )( 2 Φ n / ∂φ 2 ) Atomic orbitals of the hydrogen atom Φ n ( r , θ , φ ) are the so-called atomic orbitals . It turns out that they are products of three functions: Φ n ( r , θ , φ ) = Ψ nlm ( r , θ , φ ) = R n ( r ) Θ l ( θ ) Φ m ( φ ) n is the principal quantum number; l is the angular momentum quantum number; and m is the azimuthal quantum number. l = 0 1 2 3 4 ... n = 1 m l = 0 2 0 -1, 0, 1 3 0 -1, 0, 1 -2, -1, 0, 1, 2 4 0 -1, 0, 1 -2, -1, 0, 1, 2 -3, -2, -1, 0, 1, 2, 3 5 0 -1, 0, 1 -2, -1, 0, 1, 2 -3, -2, -1, 0, 1, 2, 3 -4, -3, -2 -1, 0, 1, 2, 3, 4 ...
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This note was uploaded on 10/27/2010 for the course BI 110 taught by Professor Richards,j during the Winter '08 term at Caltech.

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L5 - Biophysical Chemistry Chemistry 24a Winter Term...

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