b Ψ Ψ Principles of Organic Chemistry lecture 22, page 1

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1 Pinciples of Oganic Chemisty lectue, page. Basis fo LCAO and Hückel MO Theoy.. Souces... Hypephysics online. Zimmeman, H. E., Quantum Mechanics fo Oganic Chemists. ed.; Academic Pess: New Yok, NY, William H. Coppe Mathematica Compute Pogams fo Physical Chemisty, Spinge- Velag, 998.ISBN: The Opeato Postulate.. Physical obsevables have mathematical opeatos which ae used in conjunction with the wavefunction.... Wavefunctions ae mathematical expessions that completely descibe the system unde study.... Eveything chemical needs to be descibable via quantum mechanics.... Opeatos in Quantum Mechanics... Don t ty to commit these to memoy!!..3. The opeato Q acts on the wave function Ψ and gives eigan values q Fo example thee is second deivative in the time independent Hamiltonian Once this acts on the wavefunction, q in the equation below is an enegy eigan value QΨ n = q n Ψ n The expectation value postulate states that obsevable values can be had by the following: ( O ) = ( O p ) dxdydz b Ψ Ψ Ψ dxdydz above is the geneal expession The denominato is equal to when nomalized wavefunctions ae used So in the case in which nomalized wavefunctions ae used,

2 Pinciples of Oganic Chemisty lectue, page ( O ) = ( ) Ψdxdydz b Ψ O p These integals have to be taken ove all space...4. Fo the enegy of the system descibed by Ψ E = ΗΨdxdydz Ψ We ae summing enegetic contibutions of the electon in eally small volumes ove all space to get the total enegy... Applying the above to MOs.... O applying MOs to the above The Schödinge equation has two flaws.... One flaw (appoximation) is it does not account fo elativistic effects.... p (momentum) in the opeato fo kinetic enegy gets funny as p gets big.... one can think of mass as not being a constant as p inceases (as KE inceases)... The lack of elativistic effects is geatly diminished by the fact that electons have much less mass than nuclei.... Anothe flaw is it does not have an exact solution when moe than two bodies ae involved...3. Obitals ae invented to ende the Schödinge equation tactable Two electons ae assigned to each atomic obital and..3.. Molecula obitals ae composed of linea combinations of atomic obitals. Ψ = C + + C C..4. In othe wods we have to begin by assuming that is an decent way to handle the electons in the allyl system (C 3 H 5 ) We assume that the AOs ae not nomalized...5. E ( C + C + C dv = 3 3 ) ( C + C + C ) Η( C + C + C ) dv..5.. The expession above expands to a athe hideous equation due to the coss poducts You can imagine it getting a lot moe hideous if system is moe complex than allyl. E( C dv + C dv + C3 3 dv + C C dv + CC 3 3dv + CC3 ) dv = 3 Hdv + C H dv + C3 3H dv + C 3

3 Pinciples of Oganic Chemisty lectue, page 3 H dv + C C H dv + CC H dv + C C H dv CC3 H3dv + CC3 3H dv CC3.3. Lets abbeviate these tems as the following:.3.. H H dv the coulomb integal. This is the enegy of an isolated atomic obital..3.. H s H sdv the esonance integal. This is the stabilization achieved when an electon in is also allowed in adjacent s H s = H s.3.3. S dv the nomalization integal = if nomalized obitals ae used Ss sdv the ovelap integal. The extent to which and s ovelap in space..4. We can ewite the deived equation above as: E[c S + c S + c 3 S 33 + c c S + c c 3 S 3 +c c 3 S 3 ] = c H + c H + c 3 H 33 + c c H + c c 3 H 3 +c c 3 H o E[c S + c S + c 3 S 33 + c c S + c c 3 S 3 +c c 3 S 3 ] c H c H + c 3 H 33 c c H c c 3 H 3 c c 3 H The values of the integals ae fixed, detemined by molecula geomety The values of the coefficients will detemine the enegy of the molecula obitals If we diffeentiate the expession above with espect to the coefficients we should find minima fo the equation. de/dc = (H ES )c + (H ES )c + (H 3 ES 3 )c Diffeentiating with espect to c and c3 give de/dc = (H ES )c + (H ES )c + (H 3 ES 3 )c 3 de/dc 3 = (H 3 ES 3 )c + (H 3 ES 3 )c + (H 33 ES 33 )c in the case whee C, C and C 3 ae not all zeo. We can solve fo these coefficients using Came s ule.

4 Pinciples of Oganic Chemisty lectue, page 4 3 (H ES ) (H ES ) (H 3 ES 3 ) (H ES ) (H ES ) (H 3 ES 3 ) 3 (H 3 ES 3 ) (H 3 ES 3 ) (H 33 ES 33 ).4.7. This deteminant can expand to encompass moe complex molecules Futhe simplification is also possible. 3 (α E) β 0 β (α E) β 3 0 β (α E) Futhe simplification. 3 (α E)/ β 0 (α E)/ β 3 0 (α E)/ β Defining X = α-e/β = (E-α)/-β = (E-α)/ β 3 (α E)/ β 0 (α E)/ β 3 0 (α E)/ β

5 Pinciples of Oganic Chemisty lectue, page 5 3 X 0 X 3 0 X.4.8. E = α + X β When X we have taken E to equal zeo You can think of this enegy as the diffeence fom the Enegy of the lone p atomic obital Taking α = enegy of atomic p obital o 0 = enegy of atomic p obital both wok fo me.