11b - The electronic structure of atoms electron neutron...

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1 The electronic structure of atoms electron neutron proton The Hydrogen atom and Hydrogen-like ions 222 2 0 2 22 2 2 Schrodinger equation (cartesian coordin (,,) ates): 8 1 4 h mx y z Ze x xyz E xyz yz πε ψψ π ⎧⎫ ⎪⎪ = ⎨⎬ ⎡⎤ ∂∂∂ −+ + ⎢⎥ + + 2 2 0 2 2 21 sin 8s i Schrodinger equation (spherical co 1 or (, , ) (, , dinates): n ) 4 h r mrr r rE r Ze r r θ πθ ψθϕ θϕ ψ ⎛⎞ ∂∂ + + ⎜⎟ = ⎩⎭ ⎝⎠ ⎣⎦ Horribly looking equation…. . / Even more horribly looking equation…. . /// Not really… this equation can actually be solved!!! ☺☺☺☺☺☺ r v e Ze
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2 The solution N N N N ,, 24 2 222 2 n Wave functions (Hydrogen atomic orbitals): Energy leve ( 1, 2, 3 Principle quantum number (shells): 1,2,3, ,,) () (, 4, ; (The e ) E, , 8 ner ls: ll l l m nn KM N L e m I m n rR r R n rY Zem Z E h ψ θϕ θ ϕ ε = = = ×Θ ×Φ = × =− N N N N N 2 2 Angular momentum quantum number (subshells): 0,1,2,3,4, , -1 ; ( 1) /2 Magnetic quantum number (denoted in textbook): -,- 1 ,- 2 , , -2 , -1 , ; 21 v a l u e s gy only depends o o n) pd g s l z f m ml l l l ll Lm h ln L l l n h l π =… =+ + = + 2 Degeneracy 0; : The -th energy level is f 0 1, 0,1 2, -degener 1; 1, 0, 2; e 1 a ,2 t m m n m l l l m n =⇒ = =− − (same as in Bohr model!) 2 2 2 ( , , ) The probability of finding the electron between ( , , ) and ( sin ( , , ) The probability of finding the electron between (r, , ) and (r+d Normaliza ,) t r, ) , , rr xyz d xd yd z x y z x dx y dy z dz drd d dd θψ θ ϕ = + = + + ++ 2 2 2 2 22 2 2 --- 0 0 00 0 11 2 2 , ion: ( sin ( , , ) ( ) sin ( ) ( ) ( ) The probabil 1 s i n (,,) 1 ity of finding the elec l n mm nl l drr d d r drr R r d d drr R z d r r d d r r ππ θθϕ θθ ψθ ∞∞∞ × Φ = = = = ∫∫∫ ∫∫ ±²²³²²´ ±²³²´ 2 2 , tron at a distance between and from the nucleus (regardless of the direction). ( ) The probability density for finding the electron at distance r from the d r rR r + = proton. Interpretation of the radial part of the atomic orbitals
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3 Interpretation of the angular part of the atomic orbitals Specifying the values of the two angles and defines an orientation in The angular part of the atomic space. A convinient way of visu or al bitals d izing th etermines their s e orbital shape is by a hape. θϕ () () ssociating with each set of values of and a vecor that starts at the origin, whose orientation is specified by and and whose amplitude is equal to the absolute value of the angular part [ ΘΦ ]. , it is convenient to choose linear combinations of them which When the wave functions are complex Areas where the function is positive or negative can be designated by + and - , r are rea espe l. ctively . The s orbitals 3/2 0,0 0 2 3,0 00 0 0 2,0 1,0 0 11 R( ) 2 e x p ) 2 71 8 2 8 0, 0 ( ) ; ( ) 1 )2 ; (1) s orb 13 exp 2 22 e 2 xp 2 i 3 Properties: rr r r r r aa a a r a lm r r π == Θ ⎛⎞⎛ ⎞ ⎛ =− ⎜⎟⎜ ⎟ ⎜ ⎝⎠⎝ ⎠ ⎝ ⎛⎞ ⎡⎤ ⎜⎟ + ⎝⎠ ⎣⎦ ×− = ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ n,0 tals are spherical symmetric.
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This note was uploaded on 06/13/2009 for the course CHEM 260 taught by Professor Staff during the Spring '08 term at University of Michigan.

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11b - The electronic structure of atoms electron neutron...

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